Diamond cone for $\mathfrak{sl}(m,n)$

TL;DR

This paper introduces the diamond cone for the Lie superalgebra rak{sl}(m,n), using quasistandard tableaux and super jeu de taquin to establish a combinatorial basis and its compatibility with algebraic stratification.

Contribution

It defines the quasistandard tableaux and constructs the diamond cone as a combinatorial basis for the reduced shape algebra of rak{sl}(m,n), linking tableaux via super jeu de taquin.

Findings

Defined quasistandard tableaux for rak{sl}(m,n)

Established bijection between semistandard and quasistandard tableaux

Proved compatibility of the diamond cone with algebraic stratification

Abstract

In this paper, we first study the shape algebra and the reduced shape algebra for the Lie superalgebra $\mathfrak{sl}(m,n)$. We define the quasistandard tableaux, their collection is the diamond cone for $\mathfrak{sl}(m,n)$, which is a combinatorial basis for the reduced shape algebra. We realize a bijection between the set of semistandard tableaux with shape $\lambda$ and the set of quasistandard tableaux with shape $\mu\leq\lambda$, by using the 'super jeu de taquin' on skew semistandard tableaux. This gives the compatibility of the diamond cone with the natural stratification of the reduced shape algebra.