Branching rules for cell modules on a tower of the partition algebras

TL;DR

This paper develops a diagrammatic method to analyze the branching rules of cell modules in a tower of partition algebras, enabling calculation of structure constants in the Grothendieck ring.

Contribution

It introduces a novel diagrammatic approach to study branching rules and structure constants in partition algebras with non-zero parameters.

Findings

Derived explicit branching rules for cell modules

Calculated structure constants of the Grothendieck ring

Enhanced understanding of the algebraic structure of partition algebras

Abstract

Partition algebras with non-zero parameters are cellularly stratified and thus have the features of both cellular algebras and stratified algebras. Also, partition algebras form a tower of algebras. In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a tower of the partition algebras. This also allows us to calculate the structure constants of the Grothendieck ring of the tower.

Equations

$$E_M=\{(123),(1)(23),(12)(3),(13)(2),(1)(2)(3)\}.$$

$$\{(12^{\prime }{3}^{\prime }),(2),(3455^{\prime }{6}^{\prime }),(64^{\prime }),(1^{\prime })\}$$

$$\begin{minipage}{113.81102pt}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.34pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\\&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 17.77998pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 42.45998pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 67.13997pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{4}}$}}}}}}}{\hbox{\kern 91.37997pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-7.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{1^{\prime}}}$}}}}}}}{\hbox{\kern 17.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 42.01999pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{3^{\prime}}}$}}}}}}}{\hbox{\kern 66.69998pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{4^{\prime}}}$}}}}}}}{\hbox{\kern 91.37997pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}\equiv\begin{minipage}{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.34pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 17.77998pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{56.30663pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{43.94331pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{31.58pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 42.45998pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 67.13997pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{4}}$}}}}}}}{\hbox{\kern-7.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{1^{\prime}}}$}}}}}}}{\hbox{\kern 17.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{2^{\prime}}}$}}}}}}}{\hbox{\kern 42.01999pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{3^{\prime}}}$}}}}}}}{\hbox{\kern 66.69998pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{4^{\prime}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}$$

$$x=\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 99.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{88.66666pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{68.50002pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 78.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 99.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}$$

$$y=\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{67.66666pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{47.50002pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 78.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 99.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{88.66666pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{68.50002pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 78.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 99.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}$$

$$xy=\begin{minipage}[c]{116.65646pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\\&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 99.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{88.66666pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{68.50002pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 78.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 99.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern-5.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 15.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 36.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{67.66666pt}{-30.88885pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{47.50002pt}{-30.88885pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 57.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 78.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 99.5pt\raise-30.88885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-5.5pt\raise-46.33328pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 15.5pt\raise-46.33328pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 36.5pt\raise-46.33328pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 57.5pt\raise-46.33328pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{88.66666pt}{-46.33328pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{68.50002pt}{-46.33328pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 78.5pt\raise-46.33328pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 99.5pt\raise-46.33328pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}=\delta\cdot\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{25.66666pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{5.50002pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 36.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 57.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 99.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-5.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 15.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 36.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 57.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{88.66666pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{68.50002pt}{-15.44443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 78.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}{\hbox{\kern 99.5pt\raise-15.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}$$

$$\{(\rho ,S)|\rho \in E_n,|\rho |⩾r,S\text{mbox}isasubsetofthesetofallpartsof\rho \text{mbox}with|S|=r\}.$$

$$(x,S)\otimes (y,T)\otimes \pi .$$

$$\{(12^{\prime }{3}^{\prime }),(2),(3455^{\prime }{6}^{\prime }),(64^{\prime }),(1^{\prime })\}$$

$$(x,S)=([(1)(2)(345)(6)],\{(1),(345),(6)\}),$$

$$(y,T)=([(2^{\prime }{3}^{\prime })(5^{\prime }{6}^{\prime })(4^{\prime })(1^{\prime })],\{(2^{\prime }{3}^{\prime }),(4^{\prime }),(5^{\prime }{6}^{\prime })\})$$

$$\pi =(23)\in k{\Sigma }_3$$

$$\{(12^{\prime }{3}^{\prime }),(2),(3455^{\prime }{6}^{\prime }),(64^{\prime }),(1^{\prime })\}$$

$${\circ }^1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\begin{minipage}[c]{116.65646pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.78pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&&\\&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 18.65999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}}$}}}}}}}{\hbox{\kern 44.21999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}}$}}}}}}}{\hbox{\kern-7.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 18.22pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 43.78pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern-7.78pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ_{1^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 17.78pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{2^{\prime\prime}}}$}}}}}}}{\hbox{\kern 43.34pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{3^{\prime\prime}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}\equiv\delta\cdot\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}}$}}}}}}}{\hbox{\kern 40.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}~{}~{}\mathrm{in}~{}~{}V_{3}^{1}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}~{}~{}~{}~{}~{}~{}~{}(1)$$

$$~{}~{}~{}~{}~{}~{}~{}\begin{minipage}[c]{116.65646pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.78pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&&\\&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 18.65999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}}$}}}}}}}{\hbox{\kern 44.21999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}}$}}}}}}}{\hbox{\kern-7.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 18.22pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 43.78pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern-7.78pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ_{1^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{32.50665pt}{-35.16443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{20.14333pt}{-35.16443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{7.78pt}{-35.16443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 17.78pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ_{2^{\prime\prime}}}$}}}}}}}{\hbox{\kern 43.34pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{3^{\prime\prime}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}=\delta\cdot\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 16.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}}$}}}}}}}{\hbox{\kern 40.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}~{}~{}\equiv 0~{}~{}\mathrm{in}~{}~{}V_{3}^{2}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}~{}~{}~{}~{}~{}~{}~{}(2)$$

$$~{}~{}~{}~{}~{}\begin{minipage}[c]{116.65646pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.78pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&&\\&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 18.65999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}}$}}}}}}}{\hbox{\kern 44.21999pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-7.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 18.22pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 43.78pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{3^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern-7.78pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ_{1^{\prime\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{32.50665pt}{-35.16443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{20.14333pt}{-35.16443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{7.78pt}{-35.16443pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 17.78pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ_{2^{\prime\prime}}}$}}}}}}}{\hbox{\kern 43.34pt\raise-35.16443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet_{3^{\prime\prime}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}=\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{29.86668pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{18.38335pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{6.90001pt}{0.0pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 16.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{2}}$}}}}}}}{\hbox{\kern 40.70003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}}$}}}}}}}{\hbox{\kern 64.50005pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\equiv 0~{}~{}\mathrm{in}~{}~{}V_{3}^{2}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}~{}~{}~{}~{}~{}~{}~{}(3)$$

$$x(v\otimes s)=\left\{\begin{array}[]{ll}(xv)\otimes\pi(x,v)s&\text{if $xv$ has $r$ labelled dots,}\\ 0&\text{otherwise,}\end{array}\right.$$

$$\text{mbox}{\mathrm{Hom}}_{{\mathsf{P}}_n}(V_n^r\otimes M,V_n^r\otimes N)\cong \text{mbox}{\mathrm{Hom}}_{k{\Sigma }_r}(M,N).$$

$$\circ$$

$$\varphi (v\otimes m)$$

$$d\varphi (v\otimes m)$$

$$1$$

$${\circ }^1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$=\delta\cdot\begin{minipage}[c]{116.65646pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&&&&&&&&\\&&&&&&&&&&&&\\&&&&&&&&&&&&&\\&&&&&&&&&&&&&\\&&&&&&&&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 20.80002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 44.60004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.40005pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 92.20007pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 116.00009pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 135.9001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 155.80011pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 179.60013pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 203.40015pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 228.60016pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 255.20016pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 281.80017pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 303.10017pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.90001pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{239.06682pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{228.01126pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{216.9557pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{205.90015pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{194.84459pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{183.78903pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{172.73347pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{161.67792pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{150.62236pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{139.5668pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{128.51125pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{117.45569pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{106.40013pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{95.34457pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{84.28902pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{73.23346pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{62.1779pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{51.12234pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{40.06679pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{29.01123pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{17.95567pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{6.90012pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 16.90001pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@move@to{212.46681pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{201.1064pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{189.74599pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{178.38557pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{167.02516pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{155.66475pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{144.30434pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{132.94392pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{121.58351pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{110.2231pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{98.86269pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{87.50227pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{76.14186pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{64.78145pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{53.42104pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{42.06062pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}\lx@xy@move@to{30.70021pt}{-14.12444pt}{\hbox{{-}\hbox{\kern 0.0pt\raise-3.41666pt\crvi}}}}}\ignorespaces{\hbox{}}{\hbox{\kern 40.70003pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{3}}$}}}}}}}{\hbox{\kern 64.50005pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{4}}$}}}}}}}{\hbox{\kern 88.30006pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{5}}$}}}}}}}{\hbox{\kern 112.10008pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{6}}$}}}}}}}{\hbox{\kern 135.9001pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 151.9001pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{7}}$}}}}}}}{\hbox{\kern 175.70012pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\circ^{8}}$}}}}}}}{\hbox{\kern 199.50014pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{9}}$}}}}}}}{\hbox{\kern 223.30016pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{10}}$}}}}}}}{\hbox{\kern 249.90016pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{11}}$}}}}}}}{\hbox{\kern 276.50017pt\raise-14.12444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\bullet^{12}}$}}}}}}}{\hbox{\kern-3.0pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 20.80002pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 44.60004pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.40005pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 92.20007pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 116.00009pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 135.9001pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 155.80011pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 179.60013pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 203.40015pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 228.60016pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 255.20016pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 281.80017pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 303.10017pt\raise-28.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 20.80002pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 44.60004pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.40005pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 92.20007pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 116.00009pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 135.9001pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 155.80011pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 179.60013pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 203.40015pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 228.60016pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 255.20016pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 281.80017pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 303.10017pt\raise-39.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 20.80002pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 44.60004pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 68.40005pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 92.20007pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 116.00009pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 135.9001pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 155.80011pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 179.60013pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 203.40015pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 228.60016pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 255.20016pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 281.80017pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 303.10017pt\raise-50.24887pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage}$$

$$(T_{non-l},T_{label},L_{left},L_{right}),$$

$$(T_{non-l},T_{label},L_{left},L_{right})$$

$$(T_{non-l}^{\prime },T_{label}^{\prime },L_{left}^{\prime },L_{right}^{\prime })<(T_{non-l},T_{label},L_{left},L_{right})$$

Full text

††2010 Mathematics Subject Classification: 16D90;16G10;16E20.††Keywords: partition algebra; tower of algebras; cellular algebra; cell module; Grothendieck ring.

1 Introduction

To depict the Pott model in statistical mechanics, Martin [13] and Jones [10] defined the partition algebra independently in 1990. Such algebra is a typical diagram algebra, which contains the Temperley-Lieb algebra and the Brauer algebra as its subalgebras. The partition algebra is proved to be cellular [17] and in some cases is also quasi-hereditary [12]. Therefore, it admits good properties of representation and homology. Moreover, as shown in [6], the partition algebra is cellularly stratified provided the parameter is non-zero, which leads the cell modules to form a standard system and the algebra to admit a Schur algebra. Also, there is the typical Schur-Weyl duality between partition algebra and symmetric group [5].

Many diagram algebras have the following features: For same diagram algebras, the low-dimensional algebras are usually the subalgebras of high-dimensional algebras. So, we may consider using the tower of algebras to study the diagram algebras. This approach was originally used to study the tower of semi-simple algebras and called as the Jones basic construction [9]. Later, people tried to extend the framework to study the towers of non-semisimple algebras, such as the towers of quasi-hereditary and cellular algebras [2, 3].

Another important feature of the tower of diagram algebras is as follows: As two diagrams can be juxtaposed, i.e., diagram $A$ is placed on the left of diagram $B$, we can define a tensor product $A\otimes B$ of two diagrams (see Figure2(1) as an example). Thus, we can consider a tower of algebras $A_{0}\subset A_{1}\subset A_{2}\subset\cdots$ with the following axiom as a tensor product.

Axiom (T): There is an external multiplication $\rho_{m,n}:A_{m}~{}\otimes A_{n}\rightarrow A_{m+n},$ for all $m,n\geqslant 0,$ such that

(1) For all $m$ and $n$, $\rho_{m,n}$ is an injective homomorphism of algebras, sending $1_{m}\otimes 1_{n}$ to $1_{m+n}$;

(2)  $\rho$ is associative, that is, $\rho_{l+m,n}~{}\cdot~{}(\rho_{l,m}\otimes 1_{n})=\rho_{l,m+n}~{}\cdot~{}(1_{n}\otimes\rho_{m,n})$, for all $l,m,n$.

Through the tensor product, on the one hand, for any $m,n\in\mathbb{N}$, we can consider the restriction and induction functors between module categories of algebras $A_{m}~{}\otimes A_{n}$ and $A_{m+n}$; and on the other hand, the Grothendieck group of the category of finitely generated modules over a tower of algebras can be endowed with an algebra structure. In paper [16], we studied a tower of Temperley-Lieb algebras $\mathsf{TL}_{n}$ and introduced the concept of walled modules by studying the characteristics of the restricting Temperley-Lieb diagrams. Through this concept, the branching rules for cell modules of algebras $\mathsf{TL}_{m}~{}\otimes\mathsf{TL}_{n}$ and $\mathsf{TL}_{m+n}$ were obtained and the algebra structure of the Grothendieck group for the tower was calculated. For Brauer algebras, Hartmann and Paget studied permutation modules and young modules of Brauer algebras in [7], at the same time obtaining the branching rules between cell modules. However, for the partition algebra, things became more complex. Recently, Paul studied the Specht modules of partition algebras in [15] and the permutation modules of cellularly stratified algebras in [14].

In this paper, we draw on the approach used to study the tower of Temperley-Lieb algebras in [16] to study the tower of partition algebras. Precisely, In Section 2, we first recall the concept of partition algebras and the inflation approach used by Xi [17] in proving the cellularity of partition algebras. The inflation approach, introduced by König and Xi in [11], not only effectively describes the structure of cellular algebras, but is also a powerful tool for proving that an algebra admits cellularity. In [17], Xi also introduced the standard form of a partition diagram, which laid a foundation for us to study the tower of partition algebras in terms of a diagrammatic approach. We then study the diagrammatic presentation of the standard form in Section 3. In Section 4, we offer the concept of ‘walled modules’ and give the branching rules for cell modules of partition algebras $\mathsf{P}_{m}~{}\otimes\mathsf{P}_{n}$ and $\mathsf{P}_{m+n}$ based on it. In Section 5, we further compute the structure constants of the Grothendieck ring for a tower of partition algebras through the branching rules.

Throughout this paper, the partition algebras are over an algebraically closed field $k$ of arbitrary characteristic. All modules are finitely generated left modules.

2 Preliminaries

In this section, we shall recall basic definitions and facts on partition algebras, which are elementary elements in our proofs.

2.1 Partition algebra

Let $n\in\mathbb{N}$, and $M$ be a finite set. We denote by $E_{M}$ the set of all partitions of the set $M$:

$E_{M}:=\{\rho=((M_{1})(M_{2})\cdots(M_{i})\cdots)|\emptyset\neq M_{i}\subset M,~{}\cup_{i}M_{i}=M,\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}M_{i}\cap M_{j}=\emptyset~{}(i\neq j)\}$

For instance, suppose $M=\{1,2,3\}$, then

[TABLE]

Now we are interested in the case $M=\{1,...,n,1^{\prime},...,n^{\prime}\}$ with $n\in\mathbb{N}.$ Fixed an element $\delta\in k$, the partition algebra $\mathsf{P}_{n}(\delta)$ (simply denote by $\mathsf{P}_{n}$) is the associative $k$-algebra with basis consisting of the set $E_{M}$. We can also regard $\mathsf{P}_{n}(\delta)$ as a diagram algebra. This means that the basis consists of diagrams with two rows of $n$ dots each (top row labelled by $1,...,n$ and bottom row labelled by $1^{\prime},...,n^{\prime}$), where dots which belong to the same part of the partition are jointed by arcs transitively. We call such diagram an $n$-diagram. Note that this description is not unique. For example, the set partition

[TABLE]

corresponds different diagrams, but they are equivalent under the same connectivity in each part:

[TABLE]

The parts connecting top and bottom row are often called propagating parts in the literature. The number of propagating parts of an $n$-diagram is called propagating number. We call the actual line connecting a top and a bottom row dot propagating line.

The multiplication is given by concatenating two $n$-diagrams: Stacking the first diagram on top of the second diagram, identifying the bottom row of the upper with the top row of the lower diagram and following the lines from top to bottom or within one row. A part which has no dot in top or bottom row are replaced by a factor $\delta$. This multiplication is independent of the choice of diagram. The following example gives the multiplication of two $6$-diagrams.

Let

[TABLE]

and

[TABLE]

Then

[TABLE]

2.2 Standard form

We keep the notation introduced in the previous section and simply denote by $E_{n}$ the set of all partition of $\{1,2,...,n\}.$

For each $\rho\in E_{n}$, we may write $\rho$ in a standard form: Suppose $\rho=((M_{1})\cdots(M_{s}))$, we write each $M_{i}=(a_{1}^{(i)}a_{2}^{(i)}\cdots a_{t_{i}}^{(i)})$ with $a_{1}^{(i)}<a_{2}^{(i)}<\cdots<a_{t_{i}}^{(i)}$. We next define an order, called the standard order, on the set of all parts of $\rho$ by saying that $M_{j}<M_{k}$ if and only if $a_{1}^{(j)}<a_{1}^{(k)}$ and then say that $\rho$ is a standard form when $a_{1}^{(1)}<a_{1}^{(2)}<\cdots<a_{1}^{(s)}$. It is easy to see that the standard form is unique. For example, the partition $((13)(28)(456)(7))$ is the standard form.

For each $r\in\{0,1,...,n\},$ we define a $k$-space $V_{n}^{r}$ with the basis consisting of the set

[TABLE]

By such standard form, Xi [17] shows that each $\rho\in E_{M}$ can be written uniquely as an element of $V_{n}^{r}\otimes V_{n}^{r}\otimes k\Sigma_{r}$ for a number $r\in\{0,1,\cdots,n\}$, where $\Sigma_{r}$ is the symmetric group of all permutations on $\{1,2,\cdots,r\}$ and $k\Sigma_{r}$ denotes the corresponding group algebra over the field $k$.

For a partition $\rho\in E_{n}$, let $x$ be the partition of $E_{n}$ obtained by deleting all primed elements and $y$ be obtained by deleting all unprimed elements. Let $S_{\rho}$ be the set of parts of $\rho$ containing both primed and unprimed elements. We next let $S$ be the set of $x$ which are obtained from elements of $S_{\rho}$ by deleting the numbers contained in $\{1^{\prime},2^{\prime},\cdots,n^{\prime}\}$. We similarly get a subset $T$ of the set of all parts of $y$. Clearly, both $S$ and $T$ contain $r~{}(=|S_{\rho}|)$ elements.

Now we write $S=\{S_{1},\cdots,S_{r}\}$ and $T=\{T_{1},\cdots,T_{r}\}$ preserved standard order, saying, $S_{1}<S_{2}<\cdots<S_{r}$ and $T_{1}<T_{2}<\cdots<T_{r}$. Thus, we may define a permutation $\pi\in\Sigma_{r}$ by sending $j$ to $k$ if there exists a part of $S_{\rho}$ containing both $S_{j}$ and $T_{k}$. Consequently, by means of a standard form, we can associate with each $\rho$ a unique element

[TABLE]

For example, a standard form of the partition

[TABLE]

can be written

[TABLE]

[TABLE]

and

[TABLE]

Actually, Xi [17] shows that $\mathsf{P}_{n}(\delta)$ is cellular by considering it as iterated inflation of group algebras of symmetric groups, that is, there is a chain of two-sided ideals ${0}=J_{0}\subseteq J_{1}\subseteq...\subseteq J_{n}=\mathsf{P}_{n}(\delta)$ such that each subquotient $J_{r}/J_{r-1}$ ( called as layers) is a non-unital algebra of the form $V_{n}^{r}\otimes V_{n}^{r}\otimes k\Sigma_{r}$. It is easy to see that multiplication cannot move to a higher layer, i.e., for $x\in J_{r}/J_{r-1}$ and $y\in J_{s}/J_{s-1}$ we have $xy\in J_{\min\{r,s\}}$.

Theorem 2.1** ([17, Theorem 4.1 & Corollary 4.10])**

The partition algebra $\mathsf{P}_{n}(\delta)$ is cellular as an iterated inflation of the form $\bigoplus\limits_{r=0}^{n}V_{n}^{r}\otimes_{k}V_{n}^{r}\otimes_{k}k\Sigma_{r}$, with respect to the involution $i$ turning a diagram upside down. The cell modules are of the form $V_{n}^{r}\otimes_{k}v_{r}\otimes_{k}\Delta(\lambda)$ with $r\in\{0,...,n\}$ and $\lambda$ a partition of $r$, where $v_{r}$ is a fixed non-zero element of $V_{n}^{r}$ and $\Delta(\lambda)$ is a cell module for $k\Sigma_{r}$.

3 Diagrammatic presentation of the standard forms

In this section, we shall depict the characteristics of partition diagrams.

Keeping the notations in the previous section, note that, for a standard form $(x,S)\otimes(y,T)\otimes\pi$, $S$ and $T$ are in fact the parts of jointing propagating lines in the corresponding diagram, and the standard order in $S$ or $T$ is only determined by the first number in each part. Thus, the corresponding dots of these numbers become those who joint propagating lines. We hence write such dots as empty circles $\circ$ instead of dots $\bullet$, called labelled dots. Also, the parts containing labelled dots are called by labelled parts . It is clear that the labelled dots must be leftmost dots of the parts in $S$ and $T$. A diagram replaced by empty circles according to standard order is called a standard diagram.

For example, a standard form for previous partition

[TABLE]

can be presented by a standard diagram

[TABLE]

It should be noted that propagating lines can be jointed only by empty circles $\circ$. Therefore, if one top part is jointed with a bottom part, then they can be jointed only by one propagating line, for example, the empty circle $3$ in top part $(345)$ is jointed with empty circle $5^{\prime}$ in bottom part $(5^{\prime}6^{\prime})$. Hence the permutation $\pi=(23)\in\Sigma_{3}$ in previous example can be presented by

[TABLE]

Furthermore, for a standard form $(x,S)\otimes(y,T)\otimes\pi$, $(x,S)$ and $(y,T)$ can be presented by the $(n,r)$-half diagrams with $r$ labelled dots of total $n$ dots, which obtained from a standard diagram by dropping propagating lines, and the top and bottom half diagrams correspond to $(x,S)$ and $(y,T)$, respectively. For instance,

\textstyle{\circ^{1}}$$\textstyle{\bullet^{2}}$$\textstyle{\circ^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet^{5}}$$\textstyle{\circ^{6}} and

\textstyle{\bullet_{1^{\prime}}}$$\textstyle{\circ_{2^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{3^{\prime}}}$$\textstyle{\circ_{4^{\prime}}}$$\textstyle{\circ_{5^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{6^{\prime}}}

are $(6,3)$-half diagrams in previous example.

Therefore, $V_{n}^{r}$ can be also regarded as the space with the basis of all $n$ dots half diagrams with exactly $r$ labelled dots. Moreover, the space $V_{n}^{r}$ in fact admits a very natural action of the partition algebra $\mathsf{P}_{n}$ by concatenation. Precisely, for an $n$-diagram $x\in\mathsf{P}_{n}$ and an $(n,r)$-half diagrams $v\in V_{n}^{r}$, by stacking $x$ on the top of $v$ and identifying the bottom dots of $x$ with $v$, we then get a new diagram $w$ and denote its top row by $top(w)$, where a part is labelled if and only if it joined by both labelled parts of $x$ and $v$. Then we let $xv=\delta^{j}\cdot top(w)$, with $j$ the number of parts which have no dot in top row, if $w$ has exactly $r$ labelled parts(equivalently, $r$ labelled dots), and $xv=0$ otherwise.

Figure 1 illustrates that the action of an $n$-diagram $D$ and an $(n,r)$-half diagram as well as gives the following facts:

**A. If a labelled part in a half diagram is still labelled through action, then the necessary condition is that such labelled part is joined with a labelled part (in fact joined with a propagating part) in $D$. **

**B. A labelled part will vanish in the result diagram if it is joined with a non-labelled part. **

For Fact A, as shown in Figure 1(1), $(1^{\prime\prime}2^{\prime\prime})$ is a labelled part in the half diagram, $(122^{\prime})$ is a propagating part. After the two parts are concatenated, the part (12) obtained is still a labelled part. However, in Figure 1(2), $(1^{\prime\prime}3^{\prime\prime})$ is a labelled part in the half diagram and there is no propagating line concatenated with it, thus it vanishes in the result diagram. This illustrates the Fact B. Figure 1(3) shows that two labelled parts, saying $(1^{\prime\prime}3^{\prime\prime})$ and $(2^{\prime\prime})$, joined as one labelled part $(13)$ by concatenation, hence the result diagram is [math] in $V_{3}^{2}$ and the condition in Fact A is not sufficient.

We will refer to the module $V_{n}^{r}$ as the* half diagram module*. As we know, the group algebra $k\Sigma_{r}$ is cellular with cell modules the dual Specht modules $S_{\lambda}$ with $\lambda\vdash r$. By Theorem 2.1, the module $\Delta_{n}(r,\lambda)=V_{n}^{r}\otimes S_{\lambda}$ is a cell module with the action of $P_{n}$: Given an $n$-diagram $x\in P_{n}$ and a pure tensor $v\otimes s\in V_{n}^{r}\otimes S_{\lambda}$, we define

[TABLE]

where $xv$ is given above and $\pi(x,v)\in\Sigma_{r}$ is the permutation on the labelled dots of $xv$.

The following result is a generalization of [[7], Proposition 3] from Brauer algebras to partition algebras.

Proposition 3.1

Keep the notations above. Let $M$ and $N$ be $k\Sigma_{r}$-modules, and let $V_{n}^{r}$ be a half diagram module of $\mathsf{P}_{n}$. Then

[TABLE]

*Proof.  *Firstly, we denote by $v_{0}\in V_{n}^{r}$ the half diagram

[TABLE]

with $r$ labelled dots on the left hand side and $n-r$ arcs all on the right side.

For $\phi\in\mbox{\rm Hom}_{\mathsf{P}_{n}}(V_{n}^{r}\otimes M,V_{n}^{r}\otimes N)$ and $m\in M$, let $\phi(v_{0}\otimes m)=\sum_{i\in I}v_{i}\otimes n_{i}$ and apply the idempotent e_{r}=\begin{minipage}[c]{113.81102pt} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.34pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&\\&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 17.34pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 45.00659pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 66.4466pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 90.56863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 111.56863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 139.67522pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet^{n}}}}}}}}}{\hbox{\kern-7.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ_{1^{\prime}}}}}}}}}}{\hbox{\kern 17.34pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...}}}}}}}}{\hbox{\kern 45.00659pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ}}}}}}}}{\hbox{\kern 66.00659pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\circ_{r^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 90.56863pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 111.56863pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 139.23523pt\raise-17.4711pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet_{n^{\prime}}}}}}}}}}\ignorespaces}}}}\ignorespaces\end{minipage} on both sides. Note that $e_{r}(v_{0}\otimes m)=v_{0}\otimes m$ and each $e_{r}v_{i}$ is either zero or a multiple of $v_{0}$ . Hence it must be some $v_{0}\otimes n^{\prime}.$ We let $\hat{\phi}$ be the map induced on $M$ and it is clear $\hat{\phi}\in\mbox{\rm Hom}_{k\Sigma_{r}}(M,N)$. Further, it is easy to see that for each $v\in V_{n}^{r}$ there exists an element $a\in\mathsf{P}_{n}$ such that $v=av_{0}$. Therefore for each $m\in M$ and $v\in V_{n}^{r}$ we have

[TABLE]

Consequently, given any $\hat{\phi}\in\mbox{\rm Hom}_{k\Sigma_{r}}(M,N)$, we can define a may $\phi$ by previous equalities. It is further a $\mathsf{P}_{n}$-homomorphism since for $d\in\mathsf{P}_{n}$

[TABLE]

. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

4 Walled modules

In this section, we first give the concept of walled module in Definition 4.1. By using such a notion, we describe the branching rules for cell modules on a tower of the partition algebras and then give a composition series of the restricting module $\mbox{\rm Res}_{\scriptsize{\mathsf{P}_{m}\otimes\mathsf{P}_{n}}}^{\scriptsize{\mathsf{P}_{m+n}}}(\Delta_{m+n}(r,\lambda))$ in Theorem 4.2.

4.1 Walled modules

The main concept in the article comes from the following observations: Given an $m$-diagram and an $n$-diagram, we can juxtapose them into an $(m+n)$-diagram. At this point, if we imagine there is a wall between the two diagrams, this $(m+n)$-diagram will become a walled diagram, which we can call an $(m|n)$-walled diagram(simply, an $(m|n)$-diagram). It is clear that all such diagrams form a basis of $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$. Figure 2(1) demonstrates the juxtaposition of a $5$-diagram in $\mathsf{P}_{5}$ and a $4$-diagram in $\mathsf{P}_{4}$.

Similarly, when $\Delta_{m+n}(r,\lambda)=V_{m+n}^{r}\otimes S_{\lambda}$ is restricted as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, an $(m+n,r)$-half diagram in $V_{m+n}^{r}$ can be viewed as a ‘walled’ half diagram if we imagine that there is a wall which separates the dots $m,m+1$. Thus we call the diagram as an $(m|n,r)$-walled half diagram(simply, an $(m|n,r)$-half diagram) and rewrite $V_{m+n}^{r}$ by $V_{m|n}^{r}$.

An $(m|n,r)$-half diagram consists of some parts, which can be classified into the following four types:

1. Non-labelled parts crossing the wall, called Non-Labelled Through Parts;

2. Labelled parts crossing the wall, called Labelled Through Parts;

Furthermore, for such a diagram, we call all Non-Labelled Through Parts and Labelled Through Parts as *Through Parts *.

3. Non-labelled parts not crossing the wall, called Non-Labelled Normal Parts;

4. Labelled parts not crossing the wall, called Labelled Normal Parts.

Further, we call a Labelled Normal Part on the right side of the wall as a Right Labelled Normal Part, and a Labelled Normal Part on the left side as a Left Labelled Normal Part.

When an $(m|n)$-diagram $D\otimes D^{\prime}$ acts on an $(m|n,r)$-half diagram, it is easy to see the following facts:

**C. A Labelled Through Part in a walled half diagram may become Labelled Normal Part in the new walled half diagram through the action of diagram, but it will certainly not become a Non-Labelled Through Part. **

**D. A Non-Labelled Through Part in a walled half diagram may vanish in the new diagram through the action of diagram, and may also become a Labelled Through Part through a Labelled Normal Part. **

For Fact C, Figure 3 illustrates the first half of the statement. Part $(6^{\prime\prime}7^{\prime\prime})$ and $(5^{\prime\prime}8^{\prime\prime})$ are Labelled Through Parts in the half diagram and become Labelled Normal Parts, saying $(6)$ and $(8)$, by the action of diagram. The second half of the statement immediately follows form the Facts A and B: As can be known from Facts A and B, a Labelled Through Part in a half diagram may become a labelled part or vanish through action, thus it is impossible to obtain a Non-Labelled Through Part.

For Fact D, as shown in Figure 3, part $(4^{\prime\prime}9^{\prime\prime})$ is a Non-Labelled Through Part, which vanishes through the action of diagram. However, we can see that $(1^{\prime\prime}11^{\prime\prime})$ is a Non-Labelled Through Part. Through the action of diagram, it is combined with a Labelled Normal Part $(12^{\prime\prime})$ to obtain $(1~{}11)$ which is a Labelled Through Part. It is similar for $(3^{\prime\prime}10^{\prime\prime})$. The part is combined with a Labelled Normal Part $(2^{\prime\prime})$ to obtain $(2~{}10)$ which is a Labelled Through Part.

We then endow each $(m|n,r)$-half diagram with a $4$-tuple

[TABLE]

where $T_{non-l}$ is the number of the Non-Labelled Through Parts and $T_{label}$ is the number of Labelled Through Parts, $L_{left}$ and $L_{right}$ are numbers of Left and Right Labelled Normal Parts respectively. It is clear that $T_{label}+L_{left}+L_{right}=r$.

For instance, Figure 2(2) illustrates that an (17,3)-half diagram in $V_{17}^{3}$ can be viewed as a $(9|8,3)$-half diagram when $\Delta_{17}(3,\lambda)=V_{17}^{3}\otimes S_{\lambda}$ is restricted as a $\mathsf{P}_{9}\otimes\mathsf{P}_{8}$-module. Here, $(77^{\prime})$ and $(95^{\prime}4^{\prime})$ are Non-Labelled Through Parts; $(568^{\prime})$ and $(86^{\prime})$ are Labelled Through Parts; $(13)$ and $(3^{\prime}1^{\prime})$ are Left and Right Labelled Normal Parts respectively, and hence it corresponds to the tuple $(2,2,1,1)$.

We denote by $I_{m|n}^{r}$ the index set consisting of all possible tuples

[TABLE]

of the $(m|n,r)$-half diagrams, and we may further endow the index set with a partial order defined as follows: Let

[TABLE]

if

[TABLE]

or

[TABLE]

Due to the above-mentioned observations, we define

Definition 4.1

Suppose $\textbf{t}=(T_{non-l},T_{label},L_{left},L_{right})$ is a possible tuple of the $(m|n,r)$-half diagrams. Let $W_{m|n}^{\lambda}(\textbf{t})$ be the subspace $V_{m|n}^{\leqslant\textbf{t}}\otimes S_{\lambda}$ of the restriction module $\mbox{\rm Res}_{\scriptsize{\mathsf{P}_{m}\otimes\mathsf{P}_{n}}}^{\scriptsize{\mathsf{P}_{m+n}}}(\Delta_{m+n}(r,\lambda))$ with $\lambda\vdash r$, where the space $V_{m|n}^{\leqslant\textbf{t}}$ spanned by all $(m|n,r)$-half diagrams with indices in $I_{m|n}^{r}$ no more than t. We call the module $W_{m|n}^{\lambda}(\textbf{t})$ as an $(m|n)$-walled module with the tuple t.

The fact that $W_{m|n}^{\lambda}(\textbf{t})$ is actually a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module will be showed in Theorem 4.2. Through the walled modules, we then get a chain by means of a linear extension of the order on $I_{m|n}^{r}$:

[TABLE]

Now, we claim

Theorem 4.2

Keep the notation as above. Then:

(1) $W_{m|n}^{\lambda}(\textbf{t})$ is a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module;

(2) Let $\Delta_{m|n}^{\lambda}(\textbf{t})$ be a subquotient of the chain (a). Then

[TABLE]

where the space $V_{m|n}^{\textbf{t}}$ spanned by all $(m|n,r)$-half diagrams with indices t.

(3) Moreover, let $T=T_{non-l}+T_{label}$ be the number of all Through Parts. Then, as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, the summand of $\Delta_{m|n}^{\lambda}(\textbf{t})$ is filtered by cell modules of the form

[TABLE]

where we simply denote $S_{\mu_{L}}\uparrow:=\mbox{\rm Ind}_{k\Sigma_{T_{label}}}^{k\Sigma_{T_{label}+L_{left}}}(S_{\mu_{L}})$, and $\lambda_{L}\vdash L_{left},\lambda_{R}\vdash L_{right},\mu\vdash T,\mu_{L}\vdash T_{label}$ and $\mu_{N}\vdash T_{non-l}$.

Here, by Littlewood-Richardson Rule in [8], the restriction of $S_{\lambda}$ from $k\Sigma_{r}$ to $k\Sigma_{L_{left}}\otimes k\Sigma_{L_{right}}$ is filtered by modules of the form $S_{\lambda_{L}}\otimes S_{\lambda_{R}}$, denoted the multiplicity by $c^{\lambda}_{\lambda_{L}|\lambda_{R}}$, and the restriction of $S_{\mu}$ from $k\Sigma_{T}$ to $k\Sigma_{T_{label}}\otimes k\Sigma_{T_{non-l}}$ is filtered by modules of the form $S_{\mu_{L}}\otimes S_{\mu_{N}}$, denoted the multiplicity by $c^{\mu}_{\mu_{L}|\mu_{N}}$.

Therefore, the multiplicity of the form in chain (a) equals to $c^{\lambda}_{\lambda_{L}|\lambda_{R}}\cdot c^{\mu}_{\mu_{L}|\mu_{N}}$.

(4) The chain (a) after refined by (3) is a composition series of the restricting module $\mbox{\rm Res}_{\scriptsize{\mathsf{P}_{m}\otimes\mathsf{P}_{n}}}^{\scriptsize{\mathsf{P}_{m+n}}}(\Delta_{m+n}(r))$.

It is easy to see that Theorem 4.2(4) immediately follows from 4.2(3), since it is filtered by cell modules. Theorem 4.2(2) follows from the definition of walled module and 4.2(1). For proving 4.2(1) and 4.2(3), the following observations are critical.

In fact, the first observation says that an $(m|n,r)$-half diagram with a tuple $(T_{non-l},T_{label},L_{left},L_{right})$ also can be viewed as an $(m,n)$-diagram, which is a diagram with $m$ dots on the top row and $n$ dots on the bottom row marking the dots in a clockwise direction and the parts in the $(m,n)$-diagram is the same as in the $(m|n,r)$-half diagram. Figure 4 is an example. The tuple $(T_{non-l},T_{label},L_{left},L_{right})$ corresponding to the first diagram is $(2,2,1,1)$. It can be read in the second diagram as follows: $T=T_{non-l}+T_{label}=4$ is the number of propagating parts ( or propagating number); $L_{left}=1$, saying $(13)$, is the number of the top labelled part but non-propagating parts; $L_{right}=1$, saying $(1^{\prime}3^{\prime})$, is the number of the bottom labelled part but non-propagating parts; $T_{label}=2$ is the number of the labelled propagating parts, saying $(568^{\prime})$, $(86^{\prime})$, and $T_{non-l}=2$ is the number of the non-labelled propagating parts, saying $(77^{\prime})$, $(95^{\prime}4^{\prime})$.

When an $(m|n,r)$-half diagram becomes an $(m,n)$-diagram, we also can represent a propagating part in standard form. In this standard form, the dots connecting propagating strings are new labelled dots, which are denoted by $\Square$ (distinct from the old labelled dots in an $(m|n,r)$-half diagram denoted by $\circ$). Further, if a new labelled dot coincides with an old labelled dot, we denote it by $\boxcirc$. See Figure 4(2) for an example.

Another fact is that $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-action on an $(m|n,r)$-half diagram can be viewed as a left $\mathsf{P}_{m}$-right $\mathsf{P}_{n}^{op}$-action on the corresponding $(m,n)$-diagram. Here, an $n$-diagram in $\mathsf{P}_{n}^{op}$ is obtained by rotating an $n$-diagram in $\mathsf{P}_{n}$ by a half turn. Take Figure 5 as an example.

Recall that the propagating number (labelled dots) never increases in a concatenation of the two diagrams, and for an $(m|n,r)$-half diagram with a tuple $(T_{non-l},T_{label},L_{left},L_{right})$, $r(=T_{label}+L_{left}+L_{right})$ is the number of all labelled parts(or labelled dots) and $T:=T_{non-l}+T_{label}$ is the number of all Through Parts. The following lemma lists more facts of diagram action.

Lemma 4.3

Let $x$ be an $(m|n,r)$-half diagram with a tuple $(T_{non-l},T_{label},L_{left},L_{right})$, and let $d$ be an $(m|n)$-diagram. Suppose the result diagram of the $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-action $d\cdot x$ has the number $r\prime$ of labelled parts with the tuple $(T_{non-l}^{\prime},T_{label}^{\prime},L_{left}^{\prime},L_{right}^{\prime})$.Then:

(1) $T^{\prime}\leqslant T$, that is, the number of Through Parts never increases;

(2) $r\prime\leqslant r$, that is, the number of labelled parts never increases;

(3) $T_{non-l}^{\prime}\leqslant T_{non-l}$;

Suppose further $r=r^{\prime}$, then:

(4) If the number of Labelled Through Parts increases, each additional Labelled Through Part is obtained by joining a Non-Labelled Through Part with a Labelled Normal Part.

Therefore if $T_{label}\leqslant T_{label}^{\prime}$, then the number of Non-Labelled Through Parts at least decreases $T_{label}^{\prime}-T_{label}$, where

[TABLE]

(5) If there are no Non-Labelled Through Parts, then the Labelled Normal Parts cannot be converted into Labelled Through Parts. At this point, the number of Labelled Through Parts will not increase.

(6) When there are no Through Parts, the numbers of Left and right Labelled Normal Parts will remain unchanged.

*Proof.  *(1) As indicated in Figure 4, an $(m|n,r)$-half diagram can be deemed as an $(m,n)$-diagram. When the two diagrams are concatenated (can be concatenated), the propagating number will not increase.

(2) In a cell chain, the number of labelled dots of diagram in the lower layer is not more than the number of labelled dots of diagram in the higher layer.

(3) First, it can be known from (1) that the total number of Through Parts will not increase. If the number of Non-Labelled Through Parts increases, this increase is definitely obtained from the Labelled Through Parts. However, this will not happen, as can be known from Fact C.

(4) It can be obtained directly from Fact D;

(5) As there are no Non-Labelled Through Parts, a Labelled Normal Part must be joined with a Labelled Through Part if it is intended to become a Labelled Through Part in terms of the action of diagram. At this point, the number of labelled dots will decrease, which contradicts the conclusion that the number of labelled dots will remain unchanged.

(6) According to the observations above, when there are no Through Parts, an $(m|n,r)$-half diagram can be regarded as the juxtaposition (tensor) of an $m$-half diagram and an $n$-half diagram. When the diagram is in action, as the number of each side of labelled parts does not increase, the numbers of labelled parts of both $m$-half diagram and $n$-half diagram will remain unchanged in the assumption that the number of labelled parts is unchanged. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

The order defined on $I_{m|n}^{r}$ is due to the lemma and Theorem 4.2(1) holds by this order. For half diagram module $\Delta_{m|n}^{\lambda}(2,1,1,1)$ with $T_{label}+L_{left}+L_{right}=3$, Figure 6 illustrates the partial order structure on the tuples.

4.2 Proof of Theorem 4.2(3)

We are now in the position to give a proof of Theorem 4.2(3).

*Proof.  *Recall that an $(m|n,r)$-half diagram can be presented by an $(m,n)$-diagram, we can represent propagating parts in standard form. The labelled dots in the $(m|n,r)$-half diagram are called old labelled dots, denoted by $\circ$. However, those who connect propagating strings in the $(m,n)$-diagram are called new labelled dots, denoted by $\Square$. Moreover, if a new labelled dot coincides with an old labelled dot $\circ$, we denote it by $\boxcirc$.

For case $\mathbf{(T_{non-l},0,0,0)}$

In this case, there are no labelled dots in an $(m|n,r)$-half diagram, and hence $r=0$ and the walled module $\Delta_{m|n}^{\lambda}(T_{non-l},0,0,0)$ is just $V_{m|n}(T_{non-l},0,0,0)$ with $\lambda\vdash 0$. The later is a vector space spanned by a basis consisting of these walled half diagrams which just contain $T_{non-l}$ Non-Labelled Through Parts. When such a walled half diagram is presented by an $(m,n)$-diagram, an element $\sigma\in\Sigma_{T_{non-l}}$ is identified by new labelled dots $\Square$. Therefore, it can be known from the inflation theory that $V_{m|n}(T_{non-l},0,0,0)\simeq V_{m}^{T_{non-l}}\otimes V_{n}^{T_{non-l}}\otimes k\Sigma_{T_{non-l}}$ as a $k$-space. Here, $k\Sigma_{T_{non-l}}$ as a $k\Sigma_{T_{non-l}}\otimes k\Sigma_{T_{non-l}}$-module, the action is given by $(\sigma_{1}\otimes\sigma_{2})\sigma=\sigma_{1}\sigma\sigma_{2}^{-1}$. As $k\Sigma_{T_{non-l}}$ is cellular, it has a filtration with subquotients $S_{\mu}\otimes S_{\mu}$. Hence, as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, $\Delta_{m|n}^{\lambda}(T_{non-l},0,0,0)$ is filtered by modules of the form:

[TABLE]

[TABLE]

The above figure is an example, labelled dots $\Square$ on top line correspond to one $S_{\mu}$, and the labelled dots $\Square$ on bottom line correspond to another $S_{\mu}$.

For case $\mathbf{(0,T_{label},0,0)}$

In this case,

[TABLE]

and the number of total labelled parts in a walled half diagram equals $T_{label}$. According to the standard order, the old labelled dot is the leftmost dot in a labelled part, thus all old labelled dots in Labelled Through Parts are on the left of wall. These labelled dots correspond to $S_{\lambda}$. Similar to the previous case, as a walled half diagram is considered as an $(m,n)$-diagram, an element $\sigma\in\Sigma_{T_{label}}$ is decided by connecting propagating strings with new labelled dots. As the cellularity of $k\Sigma_{T_{label}}$, it has a filtration with subquotients $S_{\mu}\otimes S_{\mu}$. It should be noted that the old labelled dots on the left side of the wall coincide with all new labelled dots, hence denoted by $\boxcirc$. Thus, both $S_{\lambda}$ and one $S_{\mu}$ correspond to labelled dots $\boxcirc$ on the left side of the wall, and the other $S_{\mu}$ corresponds to labelled dots $\Square$ on the right side of the wall. Therefore, as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, $\Delta_{m|n}^{\lambda}(0,T_{label},0,0)$ is filtered by modules of the form:

[TABLE]

[TABLE]

As shown in figure above, all labelled dots $\boxcirc$ correspond to $S_{\lambda}$ and one $S_{\mu}$, and all labelled dots $\Square$ correspond to another $S_{\mu}$.

For case $\mathbf{(T_{non-l},0,L_{left},L_{right})}$

In this case, the number of labelled dots is $L_{left}+L_{right}$ with $L_{left}$ labelled dots on the left side of the wall and $L_{right}$ labelled dots on the right side. As a result, when $S_{\lambda}$ as a $k\Sigma_{L_{left}+L_{right}}$-module is restricted to be a $k\Sigma_{L_{left}}\otimes k\Sigma_{L_{right}}$-module, it has a filtration with subquotients $S_{\lambda_{L}}\otimes S_{\lambda_{R}}$, where $S_{\lambda_{L}}$ corresponds to the left labelled dots $\circ$ with $\lambda_{L}\vdash L_{left}$ and $S_{\lambda_{R}}$ corresponds to the right labelled dots $\circ$ with $\lambda_{R}\vdash L_{right}$.

Similar to case $(T_{non-l},0,0,0)$, as a walled half diagram is considered as an $(m,n)$-diagram, the $T_{non-l}$ propagating strings are jointed by new labelled dots $\Square$ and determine $k\Sigma_{T_{non-l}}$ which has a filtration with subquotients $S_{\mu}\otimes S_{\mu}$. It is noted that the set of new labelled dots $\Square$ and the set of old labelled dots $\circ$ are disjointed. Therefore, the number of total labelled dots on the top line in the $(m,n)$-diagram equals $T_{non-l}+L_{left}$. Accordingly, we consider the induction $\mbox{\rm Ind}_{k\Sigma_{T_{non-l}}\otimes k\Sigma_{L_{left}}}^{k\Sigma_{T_{non-l}+L_{left}}}(S_{\lambda_{L}}\otimes S_{\mu})$. Similarly, the number of total labelled dots on the bottom is $T_{non-l}+L_{right}$, and the induction $\mbox{\rm Ind}_{k\Sigma_{T_{non-l}}\otimes k\Sigma_{L_{rght}}}^{k\Sigma_{T_{non-l}+L_{right}}}(S_{\mu}\otimes S_{\lambda_{R}})$ is considered.

Therefore, as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, $\Delta_{m|n}^{\lambda}(T_{non-l},0,L_{left},L_{right})$ is filtered by modules of the form:

[TABLE]

[TABLE]

The figure gives an example, $S_{\lambda_{L}}$ corresponds to all dots labelled with $\circ$ on the top line, while $S_{\lambda_{R}}$ corresponds to all dots labelled with $\circ$ on the bottom line. The dots labelled with $\Square$ on the top correspond to one $S_{\mu}$, and those on the bottom correspond to another $S_{\mu}$.

For case $\mathbf{(0,T_{label},L_{left},L_{right})}$

In this case, the number of labelled dots is $T_{label}+L_{left}+L_{right}(=r)$. Precisely, the number of labelled dots on the left side of the wall is $T_{label}+L_{left}$ and the number of labelled dots on the right side is $L_{right}$. Therefore it has a filtration with subquotients $S_{\lambda_{L}}\otimes S_{\lambda_{R}}$, when $S_{\lambda}$ as a $k\Sigma_{r}$-module is restricted to be a $k\Sigma_{T_{label}+L_{left}}\otimes k\Sigma_{L_{right}}$-module with $\lambda_{L}\vdash T_{label}+L_{left},\lambda_{R}\vdash L_{right}$, and here $S_{\lambda_{L}}$ corresponds to the left labelled dots and $S_{\lambda_{R}}$ corresponds to that of the right.

Similar to Case $(0,T_{label},0,0)$, as a walled half diagram represented as an $(m,n)$-diagram, the $T_{label}$ propagating strings determine $k\Sigma_{T_{label}}$ which has a filtration with subquotients $S_{\mu}\otimes S_{\mu}$.

Also, the propagating strings determine new labelled dots. It is noted that the top old labelled dots jointing propagating strings coincide with the top new labelled dots, hence presented by $\boxcirc$, which correspond to one $S_{\mu}$. Since the number of total labelled dots on the top is $T_{label}+L_{left}$, we consider induction $S_{\mu}\uparrow:=\mbox{\rm Ind}_{k\Sigma_{T_{label}}}^{k\Sigma_{T_{label}+L_{left}}}(S_{\mu})$. Accordingly, for labelled dots on the top line we consider $S_{\lambda_{L}}\oplus(S_{\mu}\uparrow)$. However, the sets of old and new labelled dots on bottom are disjointed, so it denotes new labelled dots on bottom by $\Square$, which correspond to another $S_{\mu}$. Therefore the number of total labelled dots on bottom is $T_{label}+L_{right}$, and the induction $\mbox{\rm Ind}_{k\Sigma_{T_{label}}\otimes k\Sigma_{L_{rght}}}^{k\Sigma_{T_{label}+L_{right}}}(S_{\mu}\otimes S_{\lambda_{R}})$ is considered.

Consequently, as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, $\Delta_{m|n}^{\lambda}(0,T_{label},L_{left},L_{right})$ is filtered by modules of the form:

[TABLE]

[TABLE]

As shown in figure above, $S_{\lambda_{L}}$ corresponds to all dots labelled with $\circ$ and $\boxcirc$ on the top line, while $S_{\lambda_{R}}$ corresponds to all dots labelled with $\circ$ on bottom. The dots labelled with $\boxcirc$ on top correspond to one $S_{\mu}$, and the dots labelled with $\Square$ on bottom correspond to another $S_{\mu}$.

For case $\mathbf{(T_{non-l},T_{label},L_{left},L_{right})}$

In this case, $T(=T_{non-l}+T_{label})$ is the number of all Through Parts as well as is the number of propagating strings when a walled half diagram is considered as an $(m,n)$-diagram. The number of the old labelled dots is $T_{label}+L_{left}+L_{right}=r$ with $T_{label}+L_{left}$ old labelled dots on top and $L_{right}$ old labelled dots on bottom. As a result, it has a filtration with subquotients $S_{\lambda_{L}}\otimes S_{\lambda_{R}}$, when $S_{\lambda}$ as a $k\Sigma_{r}$-module is restricted to be a $k\Sigma_{T_{label}+L_{left}}\otimes k\Sigma_{L_{right}}$-module with $\lambda_{L}\vdash T_{label}+L_{left},\lambda_{R}\vdash L_{right}$.

The $T_{non-l}+T_{label}$ propagating strings determine $k\Sigma_{T}$ which has a filtration with subquotients $S_{\mu}\otimes S_{\mu}$, and meanwhile determine new labelled dots. Similar to the previous case, the sets of old and new labelled dots on bottom are disjointed, so it denotes new labelled dots on bottom by $\Square$, which correspond to one $S_{\mu}$. Therefore the number of total labelled dots on bottom is $T+L_{right}$, and the induction $\mbox{\rm Ind}_{k\Sigma_{T}\otimes k\Sigma_{L_{rght}}}^{k\Sigma_{T+L_{right}}}(S_{\mu}\otimes S_{\lambda_{R}})$ is considered.

However, the new labelled dots on the top line should be divided into two types: One type is those who jointing Non-Labelled Through Parts in walled diagram. Thus, they are different from the top old labelled dots when a walled diagram is considered as an $(m,n)$-diagram, hence presented by $\Square$; the other type is those who jointing Labelled Through Parts in walled diagram and they coincide with a subset of old labelled dots on top, hence presented by $\boxcirc$. Thus, according to the two types, the $k\Sigma_{T}$-module $S_{\mu}$ can be further restricted as a $k\Sigma_{T_{non-l}}\otimes k\Sigma_{T_{label}}$-module with subquotients $S_{\mu_{L}}\otimes S_{\mu_{N}}$.

Moreover, it is similar to the previous case that the set of the new labelled dots $\boxcirc$ in an $(m,n)$-diagram is a subset of all old labelled dots $\circ$ on the left side of the wall in the corresponding walled diagram. The numbers of new and old such dots are $T_{label}$ and $T_{label}+L_{left}$ respectively, we hence consider induction $S_{\mu_{L}}\uparrow:=\mbox{\rm Ind}_{k\Sigma_{T_{label}}}^{k\Sigma_{T_{label}+L_{left}}}(S_{\mu_{L}})$.

Accordingly, for the left side labelled dots of the wall we consider

[TABLE]

Consequently, as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, $\Delta_{m|n}^{\lambda}(T_{non-l},T_{label},L_{left},L_{right})$ is filtered by modules of the form:

[TABLE]

with $\lambda_{L}\vdash L_{left},\lambda_{R}\vdash L_{right},\mu\vdash T,\mu_{L}\vdash T_{label}$ and $\mu_{N}\vdash T_{non-l}$.

As indicated in Figure 4, $S_{\lambda_{L}}$ corresponds to all dots labelled with $\circ$ and $\boxcirc$ on top, while $S_{\lambda_{R}}$ corresponds to all dots labelled with $\circ$ on bottom. The dots labelled with $\boxcirc$ on top correspond to $S_{\mu_{L}}$, the dots labelled with $\Square$ on bottom correspond to $S_{\mu_{N}}$ and the dots labelled with $\Square$ on bottom correspond to another $S_{\mu}$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

5 Grothendieck ring of a tower of partition algebras

In this section, we shall recall the associative algebra structure on the Grothendieck group of the towers with Axiom (T) in 5.1. The structure constants of the Grothendieck ring over a tower of partition algebras will be calculated in 5.2.

5.1 Grothendieck ring of a tower

Let $\mathcal{A}:=(A_{n})_{n\geqslant 0}$ be a tower of algebras with Axiom (T). Denote by $G_{0}(A_{n})$ the Grothendieck group of $A_{n}$, and let $G_{0}(\mathcal{A}):=\oplus_{n\geqslant 0}G_{0}(A_{n})$. One can consider the inductions on $G_{0}(\mathcal{A})$ in terms of the tensor products.

For $m,n\in\mathbb{N}$, let $M$ be a left $A_{m}$-module, and $N$ be a left $A_{n}$-module. Recall that the tensor product $M\otimes_{k}N$ is a left $A_{m}\otimes_{k}A_{n}$-module with the action $(a\otimes b)\cdot(w\otimes u)=aw\otimes bu$ for $a\in A_{m},b\in A_{n},w\in M$ and $u\in N$.

The multiplication of the isomorphism classes of an $A_{m}$-module $M$ and an $A_{n}$-module $N$ can be defined by the induction product:

[TABLE]

where

[TABLE]

[TABLE]

for $a\in A_{m+n},~{}b\in A_{m},~{}c\in A_{n},w\in M$ and $u\in N$.

If $A_{m+n}$ is a two-sided projective $A_{m}\otimes A_{n}$-module, then the multiplication is well-defined and admits the associativity(for example, see [1, Theorem 3.5]). Thus, $G_{0}(\mathcal{A})$ is an associative algebra with $[k]$ as the unit. It is clear that the conclusion holds provided a tower $\mathcal{A}$ is semi-simple (that is, each $A_{n}$ is semi-simple ). As showed in [5], if we let $\mathbb{C}$ be the field of complex numbers, then the partition algebra $\mathsf{P}_{n}(\delta)$ is semi-simple for all but a finite number of $\delta\in\mathbb{C}$.

5.2 Structure constants

For semi-simple cases, all cell modules of the algebra $\mathsf{P}_{n}$ form a complete set of non-isomorphism simple modules by [4, Prop 3.4, 3.8]. Moreover, all tensor product of cell modules $\Delta_{m}(r_{m},\nu_{1})\otimes\Delta_{n}(r_{n},\nu_{2})$, for $r_{m},r_{n}\in\mathbb{N}$, $\nu_{1}\vdash r_{m}$ and $\nu_{2}\vdash r_{n}$, form a complete set of non-isomorphism simple modules of the semi-simple algebra $\mathsf{P}_{m}\otimes_{k}\mathsf{P}_{n}$.

Suppose $\Delta_{m+n}(r,\lambda)$ is a cell module of the partition algebra $\mathsf{P}_{m+n}$ with $\lambda\vdash r$. Recall that when $\Delta_{m+n}(r,\lambda)=V_{m+n}^{r}\otimes S_{\lambda}$ is restricted as a $\mathsf{P}_{m}\otimes_{k}\mathsf{P}_{n}$-module, each $(m+n,r)$-half diagram in $V_{m+n}^{r}$ can be viewed as a walled $(m|n,r)$-half diagram in $V_{m|n}^{r}$ as well as it can be endowed with a $4$-tuple

[TABLE]

such that $T_{label}+L_{left}+L_{right}=r.$ The index set $I_{m|n}^{r}$ consists of all possible tuples.

Recall also that $T=T_{non-l}+T_{label}$ and as a $\mathsf{P}_{m}\otimes\mathsf{P}_{n}$-module, the summand of the module $\Delta_{m|n}^{\lambda}(\textbf{t})$ is filtered by cell modules of the form

[TABLE]

Here, $\mu\vdash T=(T_{non-l}+T_{label}),\mu_{N}\vdash T_{non-l},\mu_{L}\vdash T_{label},\lambda_{L}\vdash L_{left},\lambda_{R}\vdash L_{right}.$ We simply denote as

[TABLE]

and further define

[TABLE]

Theorem 5.1

Keep the notations above. Let $\mathrm{P}(\delta)$ be a semi-simple tower of the partition algebras with $\delta\in k$, and for $m,n\in\mathbb{N}$ let $\Delta_{m}(r_{m},\nu_{1})$ and $\Delta_{n}(r_{n},\nu_{2})$ be cell modules of $\mathsf{P}_{m}$ and $\mathsf{P}_{n}$, respectively, with $r_{m},r_{n}\in\mathbb{N}$, $\nu_{1}\vdash r_{m}$ and $\nu_{2}\vdash r_{n}$. Then we have

[TABLE]

where the pair $(r,\lambda)$ in sum runs over all indices of cell modules of $\mathsf{P}_{m+n}$ and the structure constant

[TABLE]

where the index $\textbf{r}=(\mu,\mu_{N},\mu_{L},\lambda_{L},\lambda_{R})$ $(\vdash(T_{non-l},T_{label},L_{left},L_{right}))$ runs over all indices of the set $\Lambda_{m|n}^{(r,\lambda)}$ such that

[TABLE]

and the index $\omega$ runs over all partitions $\omega\vdash T_{label}+L_{left}$ and $c_{\mu_{L}}^{\omega}$ and the symbols of the form $c^{\tau}_{\tau_{1}|\tau_{2}}$ are the Littlewood-Richardson coefficients, that is, the multiplicities of the Specht module $S_{\mu_{L}}$ in restricting module $S_{\omega}\downarrow_{k\Sigma_{T_{label}}}$ and the tensor product $S_{\tau_{1}}\otimes S_{\tau_{2}}$ in restricting module $\mbox{\rm Res}(S_{\tau})$, respectively.

*Proof.  *Firstly, we have the following equalities.

[TABLE]

where

[TABLE]

Due to Theorem 4.2(3), the restricting module $\mbox{\rm Res}_{\scriptsize{\mathsf{P}_{m}\otimes\mathsf{P}_{n}}}^{\scriptsize{\mathsf{P}_{m+n}}}(\Delta_{m+n}(r,\lambda))$ is filtered by cell modules of the form $(V^{(1)}\otimes\mbox{\rm Ind}(S^{(1)}))\bigotimes(V^{(2)}\otimes\mbox{\rm Ind}(S^{(2)})):=$

[TABLE]

corresponding to the index $\textbf{r}=(\mu,\mu_{N},\mu_{L},\lambda_{L},\lambda_{R})$, and the the multiplicity equals $c^{\lambda}_{\lambda_{L}|\lambda_{R}}\cdot c^{\mu}_{\mu_{L}|\mu_{N}}.$

Recall that $\Delta_{m}(r_{m},\nu_{1})=V_{m}^{r_{m}}\otimes S_{\nu_{1}}$ and $\Delta_{n}(r_{n},\nu_{2})=V_{n}^{r_{n}}\otimes S_{\nu_{2}}$, and by Proposition 3.1, it implies $\mbox{\rm Hom}((V_{m}^{r_{m}}\otimes S_{\nu_{1}}),(V^{(1)}\otimes\mbox{\rm Ind}(S^{(1)})))$

[TABLE]

[TABLE]

and

$\mbox{\rm Hom}((V_{n}^{r_{n}}\otimes S_{\nu_{2}}),(V^{(2)}\otimes\mbox{\rm Ind}(S^{(2)})))$

[TABLE]

[TABLE]

Further, by Frobenius reciprocity, for formula (2), it yields

[TABLE]

Here, the dimension of the first summand is the Littlewood-Richardson coefficient $c^{\nu_{1}}_{\lambda_{L}|\mu_{N}}$. For the second summand, $S_{\mu_{L}}\uparrow$ can be filtrated by Specht modules $S_{\omega}$ with multiplicity $c_{\mu_{L}}^{\omega}$, and hence

[TABLE]

and the dimension equals $\sum_{\omega}c_{\mu_{L}}^{\omega}\cdot c^{\nu_{1}}_{\omega|\mu_{N}}$ by Littlewood-Richardson rule.

On the other hand, for formula (3), we have

[TABLE]

and hence the dimension is the Littlewood-Richardson coefficient $c^{\nu_{2}}_{\mu|\lambda_{R}}$.

Consequently, it yields

[TABLE]

where the index $\textbf{r}=(\mu,\mu_{N},\mu_{L},\lambda_{L},\lambda_{R})$ runs over all indices of the set $\Lambda_{m|n}^{(r,\lambda)}$ such that

[TABLE]

$\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Acknowledgement

The author is grateful to Dr. Yanbo Li and Zhankui Xiao for many helpful conversations and to Dr. Wei Hu for many useful suggestions.