A graphical categorification of the two-variable Chebyshev polynomials of the second kind

TL;DR

This paper provides a graphical categorification of two-variable Chebyshev polynomials of the second kind using the $A_2$ spider, connecting diagrammatic calculus with representation theory of $U_q( ext{sl}_3)$ and introducing a natural $q$-deformation.

Contribution

It introduces a novel diagrammatic categorification of two-variable Chebyshev polynomials via the $A_2$ spider, linking skein theory with quantum group representations.

Findings

Recursive formula satisfied by $A_2$ clasps

Definition of a $q$-deformation of the polynomials

Diagrammatic approach based on skein theory

Abstract

We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.

Equations

$${\mathcal{S}}_{(k+1,l)}(x,y)$$

$${\mathcal{S}}_{(k,l+1)}(x,y)$$

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$$P_{{+}^k}^{{+}^k}T$$

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$$

$$P_{{+}^k{-}^l}^{ϵ}={\sigma }_{{+}^k{-}^l}^{ϵ}{P}_{{+}^k{-}^l}^{{+}^k{-}^l}.$$

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\lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}} {{{}}} }{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.4477pt}{-1.75pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle{\cdots}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {}{}{}\pgfsys@moveto{22.7636pt}{-5.69089pt}\pgfsys@lineto{22.7636pt}{5.69089pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-25.60904pt}{-5.69089pt}\pgfsys@moveto{-25.60904pt}{-5.69089pt}\pgfsys@lineto{-25.60904pt}{-14.22723pt}\pgfsys@lineto{25.60904pt}{-14.22723pt}\pgfsys@lineto{25.60904pt}{-5.69089pt}\pgfsys@closepath\pgfsys@moveto{25.60904pt}{-14.22723pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.09993pt}{-12.02156pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle{\sigma^{\prime\prime}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$$

$$(P_{ϵ}^{ϵ}{)}^2$$

$$=P_{{+}^k{-}^l}^{ϵ}{P}_{{+}^k{-}^l}^{{+}^k{-}^l}(P_{{+}^k{-}^l}^{ϵ}{)}^{\ast }$$

$$=P_{{+}^k{-}^l}^{ϵ}(P_{{+}^k{-}^l}^{ϵ}{)}^{\ast }=P_{ϵ}^{ϵ}$$

$$P_{\beta }^{\gamma }{P}_{\alpha }^{\beta }$$

$$=P_{{+}^k{-}^l}^{\gamma }{P}_{{+}^k{-}^l}^{{+}^k{-}^l}(P_{{+}^k{-}^l}^{\alpha }{)}^{\ast }\text{ by Proposition refmultiprop (2)}$$

$$=P_{{+}^k{-}^l}^{\gamma }(1_{{+}^k{-}^l}+x)(P_{{+}^k{-}^l}^{\alpha }{)}^{\ast }$$

$$=P_{{+}^k{-}^l}^{\gamma }(P_{{+}^k{-}^l}^{\alpha }{)}^{\ast }=P_{\alpha }^{\gamma }.$$

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{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{2.4pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{5.19998pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-8.04814pt}{8.04814pt}\pgfsys@lineto{8.04814pt}{-8.04814pt}\pgfsys@stroke\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }{\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@moveto{-8.04814pt}{8.04814pt}\pgfsys@lineto{8.04814pt}{-8.04814pt}\pgfsys@stroke\pgfsys@invoke{ }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{-5.7904pt}{5.7904pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{3.8678pt}{-3.8678pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{-5.7904pt}{5.7904pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{3.8678pt}{-3.8678pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=c_{{+},{+}}^{*},$$

$$\displaystyle c_{{+},{-}}^{-1}=\,\leavevmode\hbox to21.3pt{\vbox to21.3pt{\pgfpicture\makeatletter\hbox{\hskip 10.64813pt\lower-10.64813pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{ {}{{}}{} {}{}{}\pgfsys@moveto{8.04814pt}{8.04814pt}\pgfsys@lineto{-8.04814pt}{-8.04814pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-0.7071}{-0.7071}{0.7071}{-0.7071}{5.154pt}{5.154pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-0.7071}{-0.7071}{0.7071}{-0.7071}{-4.5042pt}{-4.5042pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-0.7071}{-0.7071}{0.7071}{-0.7071}{5.154pt}{5.154pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-0.7071}{-0.7071}{0.7071}{-0.7071}{-4.5042pt}{-4.5042pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{2.4pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{5.19998pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-8.04814pt}{8.04814pt}\pgfsys@lineto{8.04814pt}{-8.04814pt}\pgfsys@stroke\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }{\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@moveto{-8.04814pt}{8.04814pt}\pgfsys@lineto{8.04814pt}{-8.04814pt}\pgfsys@stroke\pgfsys@invoke{ }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{-5.7904pt}{5.7904pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{3.8678pt}{-3.8678pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{-5.7904pt}{5.7904pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{3.8678pt}{-3.8678pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=c_{{+},{-}}^{*},$$

$$\displaystyle\text{(R1)}~{}\,\leavevmode\hbox to28.85pt{\vbox to33.65pt{\pgfpicture\makeatletter\hbox{\hskip 14.42723pt\lower-16.82722pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}} {}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ 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{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{8.53635pt}{-5.69089pt}\pgfsys@curveto{8.53635pt}{-7.26646pt}{7.26646pt}{-8.53635pt}{5.69089pt}{-8.53635pt}\pgfsys@curveto{1.6743pt}{-8.53635pt}{0.0pt}{-4.01659pt}{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{4.01659pt}{1.6743pt}{8.53635pt}{5.69089pt}{8.53635pt}\pgfsys@curveto{7.26646pt}{8.53635pt}{8.53635pt}{7.26646pt}{8.53635pt}{5.69089pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{2.4pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{5.19998pt}\pgfsys@invoke{ 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.4926pt}{11.58115pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle{\delta}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.54814pt}{-17.25891pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle{\bar{\delta}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.54814pt}{-17.25891pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle{\bar{\epsilon}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\in\operatorname{\mathbf{Sp}_{q}}(\delta\epsilon,\epsilon\delta),\quad c_{{\delta},{\epsilon}}^{-1}=\,\leavevmode\hbox to27.76pt{\vbox to40.37pt{\pgfpicture\makeatletter\hbox{\hskip 13.88115pt\lower-20.59192pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {}{}{}\pgfsys@moveto{-8.04814pt}{-8.04814pt}\pgfsys@lineto{8.04814pt}{8.04814pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{2.4pt}\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{5.19998pt}\pgfsys@invoke{ }{}\pgfsys@moveto{8.04814pt}{-8.04814pt}\pgfsys@lineto{-8.04814pt}{8.04814pt}\pgfsys@stroke\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }{\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ 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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.54814pt}{-17.25891pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle{\bar{\delta}}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\in\operatorname{\mathbf{Sp}_{q}}(\epsilon\delta,\delta\epsilon),$$

$$c_{\delta ,ϵ}(1_{\delta }\otimes P_{ϵ}^{ϵ})=(P_{ϵ}^{ϵ}\otimes 1_{\delta })c_{\delta ,ϵ},$$

$$c_{\delta ,ϵ}^{-1}(1_{ϵ}\otimes P_{\delta }^{\delta })=(P_{\delta }^{\delta }\otimes 1_{ϵ})c_{\delta ,ϵ}^{-1},$$

$$⟨P_{(1,1)}⟩$$

$$⟨P_{(k+1,0)}⟩$$

$$⟨P_{(k+1,l)}⟩$$

$$⟨P_{(0,l+1)}⟩$$

$$⟨P_{(k,l+1)}⟩$$

$${\iota }_1\circ p_1+{\iota }_2\circ p_2=1_Y,p_1\circ {\iota }_2=p_2\circ {\iota }_1=0,p_1\circ {\iota }_1=1_{X_1},p_2\circ {\iota }_2=1_{X_2}.$$

$$P_{+-}^{+-}+\frac{1}{[3]}{b}^{+-}{d}^{+-}=1_{+}\otimes 1_{-}.$$

$$P_{+-}^{+-}(P_{+-}^{+-}+\frac{1}{[3]}{b}^{+-}{d}_{+-})=(P_{+-}^{+-}{)}^2+\frac{1}{[3]}(P_{+-}^{+-}{b}^{+-})d_{+-}=P_{+-}^{+-}$$

$$\frac{1}{[3]}{d}_{+-}(P_{+-}^{+-}+\frac{1}{[3]}{b}^{+-}{d}_{+-})=\frac{1}{[3]}{d}_{+-}{P}_{+-}^{+-}+\frac{1}{[3{]}^2}{d}_{+-}{b}^{+-}{d}_{+-}=\frac{1}{[3]}{d}_{+-}$$

$$(P_{+-}^{+-}+\frac{1}{[3]}{b}^{+-}{d}_{+-})P_{+-}^{+-}=(P_{+-}^{+-}{)}^2+\frac{1}{[3]}(b^{+-})d_{+-}{P}_{+-}^{+-}=P_{+-}^{+-}$$

$$(P_{+-}^{+-}+\frac{1}{[3]}{b}^{+-}{d}_{+-})b^{+-}=P_{+-}^{+-}{b}^{+-}+\frac{1}{[3]}{b}^{+-}{d}_{+-}{b}^{+-}=b^{+-}.$$

Full text

This work was supported by KAKENHI (15H05739).

1. Introduction

In this paper, we give a categorification of a two-variable Chebyshev polynomial of the second kind inspired by Queffelec and Wedrich [QW18a, QW18b]. They categorified the Chebyshev polynomials and power-sum symmetric polynomials through diagrammatic categories. In [QW18a], it is shown that the Jones-Wenzl projectors satisfy the recursive formula of the Chebyshev polynomials of the second kind (resp. first kind) in the split Grothendieck group of the Karoubi envelope of the Temperley-Lieb category (resp. an affine version of the Temperley-Lieb category).

From representation theoretical point of view, many mathematician and physicist have studied multi-variable generalizations of the Chebyshev polynomials associated with root systems. For examples, Koornwinder [Koo74a, Koo74b, Koo74c, Koo74d], Hoffman and Withers [HW88] for type $A$, and [NPST10, NPT11] in general. We consider the following problem:

Question 1.1**.**

Give diagrammatic categorifications of multi-variable generalizations of the Chebyshev polynomials.

We treat with a two-variable generalization of the Chebyshev polynomials, called the $A_{2}$* Chebyshev polynomials*, of the second kind appearing in [Koo74a, Koo74b, Koo74c, Koo74d] and give a solution of the above question. The $A_{2}$ Chebyshev polynomials of the second kind is a family of two-variable polynomials $\{\mathcal{S}_{(k,l)}(x,y)\mid k,l\in\mathbb{Z}_{{}\geq 0}\}$ with integer coefficients. It is defined by the following recursive formulas [Koo74c], [NesterenkoPateraSzajewskaTereszkiewicz10]:

[TABLE]

$\mathcal{S}_{(k,l)}(x,y)=0$ if $k<0$ or $l<0$, and $\mathcal{S}_{(0,0)}(x,y)=1$.

The categorification of $\mathcal{S}_{(k,l)}$ is given by using the $A_{2}$* spider*. The $A_{2}$ spider defined in [Kup96] gives a diagrammatic description of the invariant space $\operatorname{Inv}(V_{\epsilon_{1}}\otimes V_{\epsilon_{2}}\otimes\dots\otimes V_{\epsilon_{n}})$ of the fundamental irreducible representations $V_{+}$ and $V_{-}$ of $U_{q}(\mathfrak{sl}_{3})$ and their tensor products. An invariant vector is described as a linear combination of directed uni-trivalent planar graphs with source and sink vertices. We call it $A_{2}$* web*. By using $A_{2}$ webs, we can consider the $A_{2}$ version of the Temperley-Lieb category, namely a linear category of intertwining operators between tensor powers of $V_{+}$’s and $V_{-}$’s, through $\operatorname{Hom}(A,B)\cong\operatorname{Inv}(A^{*}\otimes B)$. We call it also the $A_{2}$ spider in this paper, and denote it by $\mathbf{Sp}_{q}$. Kuperberg also defined the (internal) $A_{2}$ clasp $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ in $\operatorname{Hom}(V_{+}^{\otimes k}\otimes V_{-}^{\otimes l},V_{+}^{\otimes k}\otimes V_{-}^{\otimes l})$ for $k,l\geq 0$ as a generalization of the Joens-Wenzl idempotent. Ohtsuki and Yamada gave a recursive definition of $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ in [OY97]. Then, $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ give the $A_{2}$ Chebyshev polynomial in the following sense. Let $P_{(k,l)}$ be an object in the Karoubi envelope $\operatorname{Kar}(\mathbf{Sp}_{q})$ of $\mathbf{Sp}_{q}$ corresponding to the idempotent $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$.

Theorem 1.2**.**

The isomorphism classes of $\{P_{(k,l)}\}_{k,l\geq 0}$ satisfy the recursive formula of $\{\mathcal{S}_{(k,l)}\}_{k,l\geq 0}$ in the split Grothendieck group $K_{0}(\operatorname{Kar}(\mathbf{Sp}))$ where $\mathbf{Sp}$ is the $A_{2}$ spider at $q=1$.

In the above, we consider the $\mathbb{C}$-linear category $\mathbf{Sp}_{q}$ at $q=1$. However, we will show a reccursive formula of $P_{(k,l)}$ for the $\mathbb{C}(q^{\frac{1}{6}})$-linear category $\mathbf{Sp}_{q}$ and obtain the proof by specializing $q=1$. This recursive formula of $P_{(k,l)}$ gives a natural $q$-deformation of the $A_{2}$ Chebyshev polynomial. Many kinds of $q$-deformations of the Chebyshev polynomials have been studied in various contexts, for example, [Dup10], [Cig12a, Cig12b].

This paper is organized as follows. In Section 2, we recall definitions and properties of the $A_{2}$ spider and the $A_{2}$ clasps. They are an $A_{2}$ version of the Temperley-Lieb category and the Jones-Wenzl projectors. We categorify the $A_{2}$ Chebyshev polynomials in Section 3. All proofs are given by diagrammatic calculations in the $A_{2}$ spider.

2. The $A_{2}$ spider and the $A_{2}$ clasp

In this section, we introduce an $A_{2}$ version of the Temperley-Lieb category $\mathbf{Sp}_{q}$ based on Kuperberg’s $A_{2}$ spider [Kup96]. Our definition of $\mathbf{Sp}_{q}$ is only for proof of Theorem 1.2. Evans and Pugh studied the details about the $A_{2}$ version of the Temperley-Lieb category and the $A_{2}$ planar algebra in [EP10, EP11]. We also define the $A_{2}$ clasps which play a role analogous to the Jones-Wenzl projectors and show some fundamental properties. This section is constructed by the linear skein theory and diagrammatic calculations.

2.1. The $A_{2}$ spider

An $A_{2}$* web* is a linear combination of graphs in a disk $D=[0,1]\times[0,1]$ with marked points on $[0,1]\times\{0,1\}$ which represents an intertwining operator of $U_{q}(\mathfrak{sl}_{3})$ in manner of Reshetikhin and Turaev [RT91]. See a Turaev’s book [Tur94] and [BK01] for details. We only give a combinatorial definition of the $A_{2}$ spider which is a linear category constructed from $A_{2}$ webs. See [Kup94, Kup96] for details on relation to representation theory of $U_{q}(\mathfrak{sl}_{3})$.

Let us recall the $A_{2}$ web defined by Kuperberg [Kup96]. For any $n\in\mathbb{Z}_{{}\geq 0}$, we denote sets of marked points on $[0,1]\times\{0,1\}$ by $P_{n}=\{\,p_{1},p_{2},\dots,p_{n}\,\}$ and $Q_{n}=\{\,q_{1},q_{2},\dots,q_{n}\,\}$ where $p_{i}=(i/(n+1),0)$ and $q_{i}=(i/(n+1),1)$ for $1\leq i\leq n$. If $n=0$, then $P_{0}=Q_{0}=\emptyset$. A sign of $P_{n}$ is a map $\varepsilon_{P_{n}}\colon P_{n}\to\{{+},{-}\}$. The sign $\varepsilon_{P_{n}}$ is defined by the sequence $\varepsilon_{P_{n}(1)}\varepsilon_{P_{n}(2)}\dots\varepsilon_{P_{n}(n)}$ of ${+}$ and ${-}$. A sign of $Q_{n}$ is defined in the same way. We consider $D$ with marked points $P_{k}$ and $Q_{l}$ with signs $\varepsilon_{P_{k}}$ and $\varepsilon_{Q_{l}}$ where $k,l\in\mathbb{Z}_{{}\geq 0}$. A bipartite uni-trivalent graph $G$ in $D$ is a directed graph embedded into $D$ such that every vertex is either trivalent or univalent and the vertices are divided into sinks or sources as follows:

• •

$v$

or   {+}$$v   if $v$ is a sink,

• •

$v$

or   {-}$$v   if $v$ is a source.

An $A_{2}$* basis web* is the boundary-fixing isotopy class of a bipartite trivalent graph $G$ in $D$ such that any internal face of $D\setminus G$ has at least six sides. Let $\epsilon_{k}$ and $\epsilon_{l}$ be sequences of ${+}$ and ${-}$ with length $k$ and $l$, respectively. We denote the set of $A_{2}$ basis webs in $D$ with signed marked points $\varepsilon_{P_{k}}=\epsilon_{k}$ and $\varepsilon_{Q_{l}}=\epsilon_{l}$ by $B(\epsilon_{k},\epsilon_{l})$. The $A_{2}$* web space $W(\epsilon_{k},\epsilon_{l})$* is the $\mathbb{C}(q^{\frac{1}{6}})$-vector space spanned by $B(\epsilon_{k},\epsilon_{l})$. An $A_{2}$* web* is an element in the $A_{2}$ web space.

For example, $B{(+-+,-+-)}$ has the following $A_{2}$ basis webs:

[TABLE]

Definition 2.1** (The $A_{2}$ spider).**

The $A_{2}$* spider* $\mathbf{Sp}_{q}$ is a $\mathbb{C}(q^{\frac{1}{6}})$-linear category defined as follows.

• •

An object of $\mathbf{Sp}_{q}$ is a finite sequence of signs, that is, a map $\epsilon_{n}\colon\underline{n}\to\{{+},{-}\}$ where $\underline{n}=\{1<2<\dots<n\}$ is a finite totally ordered set for $n\in\mathbb{Z}_{{}>0}$ where the map $\epsilon_{0}$ is the map from $\underline{0}=\emptyset$. A morphism $\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{l})=\operatorname{Hom}_{\mathbf{Sp}_{q}}(\epsilon_{k},\epsilon_{l})$ is the $A_{2}$ web space $W(\bar{\epsilon}_{k},\epsilon_{l})$ where $\bar{\epsilon}_{k}$ is the opposite sign of $\epsilon_{k}$. We remark that $\mathbf{Sp}_{q}(\epsilon_{0},\epsilon_{0})$ is the $1$-dimensional vector space spanned by the empty web $D_{\emptyset}$.

• •

For $A_{2}$ basis webs $F\in\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{l})$ and $G\in\mathbf{Sp}_{q}(\epsilon_{l},\epsilon_{m})$, the composition $G\circ F=GF\in\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{m})$ is defined by gluing top side of the disk of $F$ and bottom side of the disk of $G$. If $GF$ makes $4$-, $2$-, and [math]-gons, we reduce them by

• –

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where $[n]=(q^{\frac{n}{2}}-q^{-\frac{n}{2}})/(q^{\frac{1}{2}}-q^{-\frac{1}{2}})$.

• •

The identity morphism $\mathbf{1}_{\epsilon_{k}}$ in $\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{k})$ is the $k$ parallel edges with no trivalent vertex.

• •

A tensor product of two objects is defined by their concatenation, that is, $\epsilon_{k}\otimes\epsilon_{k^{\prime}}$ is a map $\underline{k+{k^{\prime}}}\to\{{+},{-}\}$ such that $\epsilon_{k}\otimes\epsilon_{k^{\prime}}(i)=\epsilon_{k}(i)$ if $0\leq i\leq k$ and $\epsilon_{k}\otimes\epsilon_{k^{\prime}}(i)=\epsilon_{k^{\prime}}(i-k)$ if $k+1\leq i\leq k+k^{\prime}$. The tensor product of two $A_{2}$ basis webs $F\in\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{l})$ and $G\in\mathbf{Sp}_{q}(\epsilon_{k}^{\prime},\epsilon_{l}^{\prime})$ is defined by gluing right side of the disk of $F$ and left side of the disk of $G$.

We remark that an $A_{2}$ web $f\in\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{l})$ represents an intertwining operator in $\operatorname{Hom}(V_{k},V_{l})$ where $V_{k}=V_{\epsilon_{k}(1)}\otimes V_{\epsilon_{k}(2)}\otimes\dots\otimes V_{\epsilon_{k}(k)}$ and $V_{l}=V_{\epsilon_{l}(1)}\otimes V_{\epsilon_{l}(2)}\otimes\dots\otimes V_{\epsilon_{l}(l)}$. As an invariant vector $f$ is an element in $\operatorname{Inv}(V_{k}^{*}\otimes V_{l})$ and $V_{k}^{*}\cong V_{\bar{\epsilon}_{k}(k)}\otimes\dots\otimes V_{\bar{\epsilon}_{k}(2)}\otimes V_{\bar{\epsilon}_{k}(1)}$ because $V_{\pm}^{*}\cong V_{\mp}$. See [Kup96] in detail. The signs of marked points of the $A_{2}$ web space are labeled according to the signs of the invariant space.

2.2. $A_{2}$ clasps

We give a diagrammatic definition of an $A_{2}$ version of the Jones-Wenzl projectors, called the $A_{2}$ clasps, introduced in [Kup96, OY97]. The purpose of this section is to construct the general form of the $A_{2}$ clasp and to prove its properties by only using the linear skein theory.

In what follows, we omit “${\otimes}$” to describe the tensor product of objects in $\mathbf{Sp}_{q}$ and ${+}$ (resp. ${-}$) means the constant map from $\underline{1}$ to ${+}$ (resp. ${-}$). Thus, ${+}^{k}$ (resp. ${-}^{k}$) is the constant map from $\underline{k}$ to ${+}$ (resp. ${-}$) for any positive integer $k$.

Definition 2.2** (The $A_{2}$ clasp in $\mathbf{Sp}_{q}({+}^{k},{+}^{k})$).**

1. (1)

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}\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.67525pt}{-2.43054pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptstyle k-2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\quad\in\mathbf{Sp}_{q}({+}^{k},{+}^{k})

The above recursive formula defines the $A_{2}$ clasp in $\mathbf{Sp}_{q}({+}^{k},{+}^{k})$ and we denote it by $P_{{+}^{k}}^{{+}^{k}}$. The $A_{2}$ clasp in $\mathbf{Sp}_{q}({-}^{k},{-}^{k})$ is also defined by the same way and we denote it by $P_{{-}^{k}}^{{-}^{k}}$.

We introduce the following $A_{2}$ webs:

[TABLE]

[TABLE]

An $A_{2}$ basis web $B(\bar{\epsilon}_{k},\epsilon_{k})$ provides a tiling of $D$. The following Lemma is easily shown by calculating the Euler number of the tiling.

Lemma 2.3** (Ohtsuki and Yamada [OY97, Lemma 3.3]).**

For any $A_{2}$ basis web $D$ in $\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{k})$ other than $\mathbf{1}_{\epsilon_{k}}$ has $t_{{-}}^{{+}{+}}$, $t_{{+}}^{{-}{-}}$, $b^{{-}{+}}$, or $b^{{+}{-}}$ in the top side of $D$ and $t_{{-}{-}}^{{+}}$, $t_{{+}{+}}^{{-}}$, $d_{{-}{+}}$, or $d_{{+}{-}}$ in the bottom side of $D$.

Proposition 2.4** (Kuperberg [Kup96], Ohtsuki and Yamada [OY97]).**

Let $k$ be a non-negative integer, then

1. (1)

$P_{{+}^{k}}^{{+}^{k}}(\mathbf{1}_{{+}^{a}}\otimes P_{{+}^{l}}^{{+}^{l}}\otimes\mathbf{1}_{{+}^{b}})=P_{{+}^{k}}^{{+}^{k}}=(\mathbf{1}_{{+}^{a}}\otimes P_{{+}^{l}}^{{+}^{l}}\otimes\mathbf{1}_{{+}^{b}})P_{{+}^{k}}^{{+}^{k}}$ for $a+b+l=k$, 2. (2)

$P_{{+}^{k}}^{{+}^{k}}(\mathbf{1}_{{+}^{a}}\otimes t_{-}^{++}\otimes\mathbf{1}_{{+}^{b}})=0=(\mathbf{1}_{{+}^{a}}\otimes t_{++}^{-}\otimes\mathbf{1}_{{+}^{b}})P_{{+}^{k}}^{{+}^{k}}$ for $a+b+2=k$.

The above is true for the opposite sign.

Proposition 2.5** (Uniqueness).**

If a non-trivial element $T\in\mathbf{Sp}_{q}({\epsilon}^{k},{\epsilon}^{k})$ satisfies $T^{2}=T$ and Proposition 2.4 (2), then $T=P_{{\epsilon}^{k}}^{{\epsilon}^{k}}$ for $\epsilon\in\{+,-\}$.

Proof.

$P_{{+}^{k}}^{{+}^{k}}$ can be expanded as a linear sum of $A_{2}$ basis webs $P_{{+}^{k}}^{{+}^{k}}=c\mathbf{1}_{{+}^{k}}+x$ by Lemma 2.3 where $c$ is a constant and $x$ have $t_{-}^{++}$ and $t_{++}^{-}$ in the top side and in the bottom side. We can see that $c=1$ from $(P_{{+}^{k}}^{{+}^{k}})^{2}=P_{{+}^{k}}^{{+}^{k}}$ and Proposition 2.4 (2). In the same way, we know $T=\mathbf{1}_{{+}^{k}}+x^{\prime}$. Therefore,

[TABLE]

We can prove it for $\epsilon={-}$ in the same way. ∎

For an $A_{2}$ basis web $w\in B(\bar{\epsilon}_{k},\epsilon_{l})$, we define $w^{*}\in B(\bar{\epsilon}_{l},\epsilon_{k})$ as the reflection of $w$ through the horizontal line $[0,1]\times\{1/2\}$ with the opposite direcion. For the coefficient $\mathbb{C}(q^{\frac{1}{6}})$, the star operator acts on $\mathbb{C}$ by complex conjugate and $(q^{\frac{1}{6}})^{*}=q^{-\frac{1}{6}}$. In this way, we define a linear map ${}^{*}\colon\mathbf{Sp}_{q}(\epsilon_{k},\epsilon_{l})\to\mathbf{Sp}_{q}(\epsilon_{l},\epsilon_{k})$.

Remark 2.6**.**

Proposition 2.5 implies $(P_{{+}^{k}}^{{+}^{k}})^{*}=P_{{+}^{k}}^{{+}^{k}}$ and $(P_{{-}^{k}}^{{-}^{k}})^{*}=P_{{-}^{k}}^{{-}^{k}}$.

We introduce the $A_{2}$ clasp $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ in $\mathbf{Sp}_{q}({+}^{k}{-}^{l},{+}^{k}{-}^{l})$ based on [OY97].

Definition 2.7** (The $A_{2}$ clasp in $\mathbf{Sp}_{q}({+}^{k}{-}^{l},{+}^{k}{-}^{l})$).**

[TABLE]

$P_{{-}^{k}{+}^{l}}^{{-}^{k}{+}^{l}}$ is also defined by the same way.

One can prove a similar statement to Proposition 2.4.

Proposition 2.8** (Kuperberg [Kup96], Ohtsuki and Yamada [OY97]).**

Let $k$ and $l$ be non-negative integers, then

1. (1)

$P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}(\mathbf{1}_{{+}^{a}}\otimes P_{{+}^{s}{-}^{t}}^{{+}^{s}{-}^{t}}\otimes\mathbf{1}_{{-}^{b}})=P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}=(\mathbf{1}_{{+}^{a}}\otimes P_{{+}^{s}{-}^{t}}^{{+}^{s}{-}^{t}}\otimes\mathbf{1}_{{-}^{b}})P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ for $a+s=k$ and $b+t=l$, 2. (2)

$P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}(\mathbf{1}_{{+}^{a}}\otimes b^{+-}\otimes\mathbf{1}_{{-}^{b}})=0=(\mathbf{1}_{{+}^{a}}\otimes d_{+-}\otimes\mathbf{1}_{{-}^{b}})P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ for $a+1=k$ and $b+1=l$.

The same equalities hold for $P_{{-}^{k}{+}^{l}}^{{-}^{k}{+}^{l}}$.

One can prove the uniqueness of $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ in a similar way to Proposition 2.5.

Proposition 2.9** (Uniqueness).**

A non-trivial idempotent element in $\mathbf{Sp}_{q}({+}^{k}{-}^{l},{+}^{k}{-}^{l})$ satisfying Proposition 2.8 (2) is uniquely determined.

Proof.

In the same way as the proof of Proposition 2.5. ∎

Remark 2.10**.**

Proposition 2.9 implies $(P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})^{*}=P_{{+}^{k}{{-}^{l}}}^{{+}^{k}{-}^{l}}$ and $(P_{{-}^{k}{+}^{l}}^{{-}^{k}{+}^{l}})^{*}=P_{{-}^{k}{+}^{l}}^{{-}^{k}{+}^{l}}$.

We give an explicit definition of a general form of the $A_{2}$ clasp appear in Kuperberg [Kup96] and Kim [Kim07]. This $A_{2}$ clasp is an $A_{2}$ web, no longer idempotent, in $\mathbf{Sp}_{q}(\epsilon,\delta)$ such that $k=\#\epsilon^{-1}({+})=\#{\delta}^{-1}({+})$ and $l=\#\epsilon^{-1}({-})=\#{\delta}^{-1}({-})$. We introduce the following $A_{2}$ basis webs:

[TABLE]

Let $k$ and $l$ be non-negative integers. We take an arbitrary object $\epsilon\colon\underline{k+l}\to\{\pm\}$ in $\mathbf{Sp}_{q}$ satisfying $k=\#\epsilon^{-1}(+)$ and $l=\#\epsilon^{-1}(-)$. Then, we define an $A_{2}$ basis web $\sigma_{{+}^{k}{-}^{l}}^{\epsilon}$ as follows. We consider the disk $D=[0,1]\times[0,1]$ with marked points signed by ${+}^{k}{-}^{l}$ and $\epsilon$. Join the marked points labeled by ${+}$ in the bottom side with ones of the upper side by straight arcs. In the same way, join the mark points labeled by ${-}$ by straight arcs. Then, one can obtain the $A_{2}$ basis web $\sigma_{{+}^{k}{-}^{l}}^{\epsilon}$ by replacing all crossing points by $H_{{+}{-}}^{{-}{+}}$.

Definition 2.11**.**

Let $k$ and $l$ be non-negative integers. Then, an $A_{2}$ clasp $P_{{+}^{k}{-}^{l}}^{\epsilon}$ in $\mathbf{Sp}_{q}({{+}^{k}{-}^{l}},{\epsilon})$ is defined by

[TABLE]

Proposition 2.12**.**

Compositions of $P_{{+}^{k}{-}^{l}}^{\epsilon}$ with $A_{2}$ basis webs $t$ and $d$ vanish.

Proof.

We prove it by induction on the number $h(\sigma)$ of $H_{{+}{-}}^{{-}{+}}$ contained in $\sigma=\sigma_{{+}^{k}{-}^{l}}^{\epsilon}$. If $h(\sigma)=0$, it is clear since $P_{{+}^{k}{-}^{l}}^{\epsilon}=P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ and Proposition 2.8. If $h(\sigma)=1$, then $\sigma=\mathbf{1}_{{+}^{k-1}}\otimes H_{{+}{-}}^{{-}{+}}\mathbf{1}_{{-}^{l-1}}$. One can prove by easy calculations. When $h(\sigma)=n+1$ $(n\geq 1)$, $\sigma$ is described as a composition of $\sigma^{\prime}$ with $\mathbf{1}_{\alpha}\otimes H_{{+}{-}}^{{-}{+}}\otimes\mathbf{1}_{\beta}$ whrere $\epsilon=\alpha{-}{+}\beta$ and $\sigma^{\prime}=\sigma_{{+}^{k}{-}^{l}}^{\alpha{+}{-}\beta}$ such that $h(\sigma^{\prime})=n$. Thus, $\sigma$ is   \scriptstyle{\cdots}$$\scriptstyle{\cdots}$$\scriptstyle{\sigma^{\prime}}   ,   \scriptstyle{\cdots}$$\scriptstyle{\cdots}$$\scriptstyle{\sigma^{\prime}}   ,   \scriptstyle{\cdots}$$\scriptstyle{\cdots}$$\scriptstyle{\sigma^{\prime}}   or   \scriptstyle{\cdots}$$\scriptstyle{\cdots}$$\scriptstyle{\sigma^{\prime}}   . In the first case, one can show by easy calculations and the induction hypothesis. 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{}{}{}\pgfsys@moveto{22.7636pt}{-5.69089pt}\pgfsys@lineto{22.7636pt}{5.69089pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-25.60904pt}{-5.69089pt}\pgfsys@moveto{-25.60904pt}{-5.69089pt}\pgfsys@lineto{-25.60904pt}{-14.22723pt}\pgfsys@lineto{25.60904pt}{-14.22723pt}\pgfsys@lineto{25.60904pt}{-5.69089pt}\pgfsys@closepath\pgfsys@moveto{25.60904pt}{-14.22723pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.54994pt}{-12.02156pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptstyle{\sigma^{\prime}}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {}{}{}\pgfsys@moveto{8.53635pt}{5.69089pt}\pgfsys@lineto{8.53635pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{8.53635pt}{7.17633pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}=0. By construction of $\sigma$, the right leg of $H_{{+}{-}}^{{-}{+}}$ and the up-pointing arc on one’s right should have a crossing ($H_{{+}{-}}^{{-}{+}}$). Then, there exists $\sigma^{\prime\prime}$ such that $h(\sigma^{\prime\prime})=n-1$ and

[TABLE]

These terms has $t_{{+}{+}}^{-}$ and $d_{{-}{+}}$ on the web $\sigma^{\prime\prime}$. 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}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}=0, by the induction hypothesis for $\sigma^{\prime\prime}\circ P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$. For other cases, we can prove in the same way. ∎

Proposition 2.13**.**

Let us decompose $\epsilon$ into three subsequences $\alpha\beta\gamma$ such that $\beta$ is expressed as the form ${{+}{+}\cdots{+}{-}{-}\cdots{-}}$ or ${{-}{-}\cdots{-}{+}{+}\cdots{+}}$. Then,

1. (1)

$P_{{+}^{k}{-}^{l}}^{\epsilon}=(\mathbf{1}_{\alpha}\otimes P_{\beta}^{\beta}\otimes\mathbf{1}_{\gamma})P_{{+}^{k}{-}^{l}}^{\epsilon}$, 2. (2)

$(P_{{+}^{k}{-}^{l}}^{\epsilon})^{*}P_{{+}^{k}{-}^{l}}^{\epsilon}=P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$.

Especially, $tP_{{+}^{k}{-}^{l}}^{\epsilon}=0$ and $dP_{{+}^{k}{-}^{l}}^{\epsilon}=0$ where $t$ (resp. $d$) is a tensor product of identity morphisms and $t_{{-}{-}}^{{+}}$ or $t_{{+}{+}}^{{-}}$ (resp. $d_{{+}{-}}$ or $d_{{-}{+}}$).

Proof.

(1) is easily shown by expanding $P_{\beta}^{\beta}$. We can desctibe $P_{{+}^{k}{-}^{l}}^{\epsilon}$ as a product $\tau P_{{+}^{k}{-}^{l}}^{\epsilon^{\prime}}$ where $\tau$ is a tensor product of identity morphisms and only one $H_{{+}{-}}^{{-}{+}}$ or $H_{{-}{+}}^{{+}{-}}$. Then, $(P_{{+}^{k}{-}^{l}}^{\epsilon})^{*}P_{{+}^{k}{-}^{l}}^{\epsilon}=(P_{{+}^{k}{-}^{l}}^{\epsilon^{\prime}})^{*}\tau^{*}\tau P_{{+}^{k}{-}^{l}}^{\epsilon^{\prime}}$ and one can finish the proof of (2) by applying the defining relation of the $A_{2}$ web to $\tau^{*}\tau$. ∎

Let us define an $A_{2}$ clasp $P_{\epsilon}^{\epsilon}$ in $\mathbf{Sp}_{q}(\epsilon,\epsilon)$ by $P_{{+}^{k}{-}^{l}}^{\epsilon}(P_{{+}^{k}{-}^{l}}^{\epsilon})^{*}$. Then,

Proposition 2.14**.**

1. (1)

$(P_{\epsilon}^{\epsilon})^{2}=P_{\epsilon}^{\epsilon}$, 2. (2)

$tP_{\epsilon}^{\epsilon}$, $dP_{\epsilon}^{\epsilon}$, $P_{\epsilon}^{\epsilon}b$, and $P_{\epsilon}^{\epsilon}t^{\prime}$ vanish.

In the above, $t^{\prime}$ (resp. $b$) is a tensor product of identity morphisms and $t_{{-}}^{{+}{+}}$ or $t_{{+}}^{{-}{-}}$ (resp. $b^{{-}{+}}$ or $b^{{+}{-}}$).

Proof.

[TABLE]

∎

Proposition 2.15** (Uniqueness).**

If $T$ in $\mathbf{Sp}_{q}(\epsilon,\epsilon)$ satisfies the conditions (1) and (2) of Proposition 2.14, then $T=P_{\epsilon}^{\epsilon}$.

Proof.

In the same way as the proof of Proposition 2.5. ∎

Remark 2.16**.**

Proposition 2.15 implies $(P_{\epsilon}^{\epsilon})^{*}=P_{\epsilon}^{\epsilon}$.

For any non-negative integers $k$ and $l$, we consider a subset $\mathbf{Sp}_{q}^{(k,l)}=\{\,\epsilon\in\mathbf{Sp}_{q}\mid k=\#\epsilon^{-1}({+}),l=\#\epsilon^{-1}({-})\,\}$ of $\mathbf{Sp}_{q}$.

Proposition 2.17**.**

There exist a set of $A_{2}$ webs $\{\,P_{\alpha}^{\beta}\in\mathbf{Sp}_{q}(\alpha,\beta)\mid\alpha,\beta\in\mathbf{Sp}_{q}^{(k,l)}\,\}$ satisfying

1. (1)

$P_{\beta}^{\gamma}P_{\alpha}^{\beta}=P_{\alpha}^{\gamma}$ for any $\alpha,\beta,\gamma\in\mathbf{Sp}_{q}^{(k,l)}$, 2. (2)

$tP_{\alpha}^{\beta}$, $dP_{\alpha}^{\beta}$, $P_{\alpha}^{\beta}b$, and $P_{\alpha}^{\beta}t^{\prime}$ vanish for any $\alpha,\beta\in\mathbf{Sp}_{q}^{(k,l)}$.

Proof.

Let us define $P_{\alpha}^{\beta}=P_{{+}^{k}{-}^{l}}^{\beta}(P_{{+}^{k}{-}^{l}}^{\alpha})^{*}$ for any $\alpha,\beta,\gamma\in\mathbf{Sp}_{q}^{(k,l)}$. Then, it is obvious that $P_{\alpha}^{\beta}$ satisfies (2) because of Proposition 2.13 (1). We show the equation of (1):

[TABLE]

∎

2.3. Braidings in $\mathbf{Sp}_{q}$

We introduce A braiding $\{c_{\delta,\epsilon}\colon\delta\otimes\epsilon\to\epsilon\otimes\delta\}$ which is a family of isomorphisms in $\mathbf{Sp}_{q}$. A definition of the braiding in the diagrammatic category $\mathbf{Sp}_{q}$ is given by Kuperberg [Kup94, Kup96]. In detail about a general theory of braidings in monoidal categories, for example, see [Tur91].

Let us define a description of an $A_{2}$ web by using a crossing with over/under information introduced in Kuperberg [Kup96]:

[TABLE]

We also describe their inverses as

[TABLE]

By the above description, we can consider $A_{2}$ webs with over/under crossings. These $A_{2}$ webs satisfies the Reidemeister moves (R1)–(R4) for framed tangled trivalent graphs, that is, we can confirm the following local moves of $A_{2}$ webs:

[TABLE]

For any objects $\delta$ and $\epsilon$ in $\mathbf{Sp}_{q}$, we define

[TABLE]

where an edge labeled by $\delta$ (resp. $\epsilon$) mean an embedding of $\mathbf{1}_{\delta}$ (resp. $\mathbf{1}_{\epsilon}$) along it with the same over/under information at every crossings. The invariance under the Reidemeister moves (R1)–(R4) provides the invariance under (R1)–(R3) for any labeled edges. By the same reason, we can slide $A_{2}$ clasps across over/under crossings, namely,

[TABLE]

3. A categorification of $\mathcal{S}_{(k,l)}(x,y)$

Let us briefly recall the definition of the Karoubi envelope and the split Grothendieck group.

Definition 3.1** (The Karoubi envelope).**

The Karoubi envelope of a category $\mathcal{A}$, denoted by $\operatorname{Kar}(\mathcal{A})$, is defined as follows:

• •

Objects in $\operatorname{Kar}(\mathcal{A})$ is pairs $(X,f)$ of objects $X$ in $\mathcal{A}$ and idempotents $f\in\operatorname{Hom}_{\mathcal{A}}(X,X)$.

• •

Morphisms in $\operatorname{Hom}_{\operatorname{Kar}(\mathcal{A})}((X,f),(Y,g))$ is morphisms $\phi\in\operatorname{Hom}_{\mathcal{A}}(X,Y)$ satisfying $g\circ\phi\circ f=\phi$ for any objexts $(X,f)$ and $(Y,g)$.

We remark that the identity in $\operatorname{Hom}_{\operatorname{Kar}(\mathcal{A})}((X,f),(X,f))$ is given by $f$. If $\mathcal{A}$ is monoidal, then the Karoubi envelope $\operatorname{Kar}(\mathcal{A})$ inherits a tensor product with $(X,f)\otimes(Y,g)=(X\otimes Y,f\otimes g)$.

Definition 3.2** (The split Grothendieck group).**

The split Grothendieck group of an additive category $\mathcal{A}$ with $\oplus$ is the abelian group $K_{0}(\mathcal{A})$ generated by isomorphism classes $\langle X\rangle$ of objects $X$ in $\mathcal{A}$ modulo the relations $\langle X_{1}\oplus X_{2}\rangle=\langle X_{1}\rangle+\langle X_{2}\rangle$. If $\mathcal{A}$ is monoidal, that is $\mathcal{A}$ equipped with a tensor $\otimes$ and the identity object $e$, then $K_{0}(\mathcal{A})$ inherits a ring structure with the unit $\langle e\rangle$ and the multiplication $\langle X\otimes Y\rangle=\langle X\rangle\cdot\langle Y\rangle$.

In this section, we consider the pairs $\{(\epsilon,P_{\epsilon}^{\epsilon})\}$ in the Karoubi envelope $\operatorname{Kar}(\mathbf{Sp}_{q})$ of the $A_{2}$ spider and its split Grothendieck group. Firstly, it is easy to see that we can take a standard representatives for the isomorphism class of $\{(\epsilon,P_{\epsilon}^{\epsilon})\}$.

Lemma 3.3**.**

For any $\epsilon\in\mathbf{Sp}_{q}^{(k,l)}$, $(\epsilon,P_{\epsilon}^{\epsilon})\cong({+}^{k}{-}^{l},P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})$ in $\operatorname{Kar}(\mathbf{Sp}_{q})$.

Proof.

$P_{\epsilon}^{{+}^{k}{-}^{l}}$ and $P_{{+}^{k}{-}^{l}}^{\epsilon}$ give an isomorphism between $(\epsilon,P_{\epsilon}^{\epsilon})$ and $({+}^{k}{-}^{l},P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})$. In fact, Proposition 2.17 shows $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}P_{\epsilon}^{{+}^{k}{-}^{l}}P_{\epsilon}^{\epsilon}=P_{\epsilon}^{{+}^{k}{-}^{l}}$ and $P_{\epsilon}^{\epsilon}P_{{+}^{k}{-}^{l}}^{\epsilon}P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}=P_{{+}^{k}{-}^{l}}^{\epsilon}$. Thus, $P_{\epsilon}^{{+}^{k}{-}^{l}}$ and $P_{{+}^{k}{-}^{l}}^{\epsilon}$ are mophisms in $\operatorname{Kar}(\mathbf{Sp}_{q})$. By the same reason, $P_{{+}^{k}{-}^{l}}^{\epsilon}P_{\epsilon}^{{+}^{k}{-}^{l}}=P_{\epsilon}^{\epsilon}$ and $P_{\epsilon}^{{+}^{k}{-}^{l}}P_{{+}^{k}{-}^{l}}^{\epsilon}=P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$. ∎

The multiplication in $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$ is commutative because of a property (2.3) of a braiding.

Lemma 3.4**.**

For any objects $\epsilon$ and $\delta$ in $\mathbf{Sp}_{q}$, $P_{\epsilon}^{\epsilon}\otimes P_{\delta}^{\delta}\cong P_{\delta}^{\delta}\otimes P_{\epsilon}^{\epsilon}$ by $c_{\epsilon,\delta}$.

Let us denote objects $({+}^{k}{-}^{l},P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})$ in $\operatorname{Kar}(\mathbf{Sp}_{q})$ by $P_{(k,l)}$ for $k,l\geq 1$. We denote the pair $(\epsilon_{0},D_{\emptyset})$ of the empty sign and the empty web by $P_{(0,0)}$. The split Grothendieck group $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$ is a ring with with the unit $1=\langle P_{(0,0)}\rangle$. We denote $\langle({+},\mathbf{1}_{+})\rangle$ and $\langle({-},\mathbf{1}_{-})\rangle$ by $X$ and $Y$ in $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$, respectively.

We will show the isomorphism classes $\langle P_{(k,l)}\rangle$ satisfy the recursive formula of the $A_{2}$ Chebyshev polynomials in $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$:

[TABLE]

We only have to prove (3.1)–(3.14) because of $(P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})^{*}\cong P_{{-}^{k}{+}^{l}}^{{-}^{k}{+}^{l}}$.

We use the following well-known fact about an additive category, see [ML98, KS06], for example.

Lemma 3.5**.**

Let $\mathcal{A}$ be an additive category and $X_{1},X_{2},Y$ objects in $\mathcal{A}$. $Y\cong X_{1}\oplus X_{2}$ if and only if there exists morphisms $p_{1}\colon Y\to X_{1}$, $p_{2}\colon Y\to X_{2}$, $\iota_{1}\colon X_{1}\to Y$, and $\iota_{2}\colon X_{2}\to Y$ satisfying the following conditions:

[TABLE]

Proposition 3.6**.**

$\langle P_{(1,1)}\rangle=XY-1$

Proof.

By Definition 2.7,

[TABLE]

Let us prove $({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})\cong P_{(1,1)}\oplus P_{(0,0)}$. It is only necessary to construct projections $p_{1}\colon({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})\to P_{(1,1)}$ and $p_{2}\colon({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})\to P_{(0,0)}$, and inclusions $\iota_{1}\colon P_{(1,1)}\to({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})$ and $\iota_{2}\colon P_{(0,0)}\to({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})$ satisfying the conditions in Lemma 3.5. These projections are given by $p_{1}=P_{{+}{-}}^{{+}{-}}$ and $p_{2}=\frac{1}{[3]}d_{{+}{-}}$ because

[TABLE]

by Proposition 2.8, and

[TABLE]

by Proposition 2.8 and d_{{+}{-}}b^{{+}{-}}=\,\leavevmode\hbox to10.64pt{\vbox to10.64pt{\pgfpicture\makeatletter\hbox{\hskip 5.32158pt\lower-5.32158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}} {}{}{}{}{}{}{}{} }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{5.12158pt}{0.0pt}\pgfsys@curveto{5.12158pt}{2.8286pt}{2.8286pt}{5.12158pt}{0.0pt}{5.12158pt}\pgfsys@curveto{-2.8286pt}{5.12158pt}{-5.12158pt}{2.8286pt}{-5.12158pt}{0.0pt}\pgfsys@curveto{-5.12158pt}{-2.8286pt}{-2.8286pt}{-5.12158pt}{0.0pt}{-5.12158pt}\pgfsys@curveto{2.8286pt}{-5.12158pt}{5.12158pt}{-2.8286pt}{5.12158pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{{}{}}{{}{}}{{}{}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-0.00179}{-1.0}{1.0}{-0.00179}{-5.11911pt}{1.36926pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{}{}}{{}{}}{{}{}}{{}{}{}{}{{}}{}{{}}}{{}{}{}{}{{}}{}{{}}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{{}{}{}{}{{}}{}{{}} {{{}}} }{{}{}}{{}{}}{}{}{{}}{{}}{{}{}}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=[3]. These inclusions are given by $\iota_{1}=P_{{+}{-}}^{{+}{-}}$ and $\iota_{2}=b^{{+}{-}}$ because

[TABLE]

and

[TABLE]

In fact, it is easily to see that

[TABLE]

Thus, we obtain $P_{(1,1)}\oplus P_{(0,0)}\cong({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})$. In $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$, it is interpreted as $\langle P_{(1,1)}\rangle+1=\langle({+}{-},P_{{+}{-}}^{{+}{-}}+\frac{1}{[3]}b^{{+}{-}}d_{{+}{-}})\rangle=XY$. ∎

Proposition 3.7**.**

For any positive integer $k$, $\langle P_{(k+1,0)}\rangle=X\langle P_{(k,0)}\rangle-\langle P_{(k-1,1)}\rangle$.

Proof.

By Definition 2.2,

[TABLE]

Let us denote the LHS of (3.6) by $f$. We show that the pair $({+}^{k+1},f)$ is isomorphic to the direct sum $P_{(k+1,0)}\oplus P_{(k-1,1)}$ in $\operatorname{Kar}(\mathbf{Sp}_{q})$. We only have to construct projections $p_{1}\colon({+}^{k+1},f)\to P_{(k+1,0)}$ and $p_{2}\colon({+}^{k+1},f)\to P_{(k-1,1)}$, and inclusions $\iota_{1}\colon P_{(k+1,0)}\to({+}^{k+1},f)$ and $\iota_{2}\colon P_{(k-1,1)}\to({+}^{k+1},f)$ satisfying (3.5). We confirm that

[TABLE]

provide projections and

[TABLE]

inclusions. It is easy to see that $p_{1}$ and $\iota_{1}$ is a morphism between $({+}^{k+1},f)$ and $P_{(k+1,0)}$. We show that $p_{2}$ is a morphism in $\operatorname{Kar}(\mathbf{Sp}_{q})$, that is,

[TABLE]

By Proposition 2.8, the LHS of the above equation is

[TABLE]

We diagrammatically calculate it using Definition 2.2:

[TABLE]

The third equation uses $[2][k]=[k+1]+[k-1]$. To prove $\iota_{2}$ is a morphism in $\mathbf{Sp}_{q}$, we have to compute $f(P_{{+}^{k}}^{{+}^{k}}\otimes\mathbf{1}_{+})(\mathbf{1}_{{+}^{k-1}}\otimes t_{{-}}^{{+}{+}})P_{{+}^{k-1}{-}}^{{+}^{k-1}{-}}P_{{+}^{k-1}{-}}^{{+}^{k-1}{-}}$. By Proposition 2.8, it reduce to

[TABLE]

We can calculate it by turning the diagrams in (3.7) upside down. Moreover, it can be confirmed that $\iota_{2}\circ p_{2}=\frac{[k]}{[k+1]}(P_{{+}^{k}}^{{+}^{k}}\otimes\mathbf{1}_{+})(\mathbf{1}_{{+}^{k-1}}\otimes t_{{-}}^{{+}{+}}t_{{+}{+}}^{-})(P_{{+}^{k}}^{{+}^{k}}\otimes\mathbf{1}_{+})=P_{{+}^{k}}^{{+}^{k}}\otimes\mathbf{1}_{+}$ by Definition 2.7 and Poroposition 2.8 and $p_{2}\circ\iota_{2}=P_{{+}^{k-1}{-}}^{{+}^{k-1}{-}}$ by inserting $P_{{+}^{k-1}{-}}^{{+}^{k-1}{-}}$ into the diagrams of (3.7). Therefore, these projections and inclusions satisfy the condition in Lemma 3.5 and give $({+}^{k+1},f)\cong P_{(k+1,0)}\oplus P_{(k-1,1)}$ in $\operatorname{Kar}(\mathbf{Sp}_{q})$. In terms of $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$,

[TABLE]

∎

Let us prepare some lemmata to prove (3.14). We introduce an $A_{2}$ basis web represented by a rectangle with a diagonal line which consist of $H_{{+}{-}}^{{-}{+}}$ and $H_{{-}{+}}^{{{+}{-}}}$.

Definition 3.8**.**

Let $m$ and $n$ be positive integers.

1. (1)

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{}{{}{}}{{}{}}{}{}{}{{}}{{}}{{}{}}{}{}{{}}{{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-11.38179pt}{4.33089pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} {}{{}}{} {}{} {}{}{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@lineto{11.38179pt}{0.0pt}\pgfsys@lineto{11.38179pt}{-8.53635pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{{}{}}{}{}{}{{}}{{}}{{}{}}{}{}{{}}{{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{11.38179pt}{-5.23088pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}} {{{}}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{{}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} {}{{}}{} {}{}{}\pgfsys@moveto{17.0727pt}{-8.53635pt}\pgfsys@lineto{17.0727pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{17.0727pt}{-1.36pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope 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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.3227pt}{12.06935pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptstyle 1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\in\mathbf{Sp}_{q}({+}^{n}{-}^{m},{-}^{m}{+}^{n})

Lemma 3.9** (Kim [Kim07, Proposition 3.1]).**

For any positive integer $k$,

[TABLE]

Lemma 3.10**.**

Let $k$ be a positive integer and X(k;i)=\,\leavevmode\hbox to45.75pt{\vbox to53.63pt{\pgfpicture\makeatletter\hbox{\hskip 10.86343pt\lower-21.41553pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{ {}{{}}{} {}{}{}\pgfsys@moveto{-5.69089pt}{-9.10509pt}\pgfsys@lineto{-5.69089pt}{20.48645pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}\pgfsys@moveto{2.84544pt}{2.27626pt}\pgfsys@lineto{2.84544pt}{2.82881pt}\pgfsys@curveto{2.84544pt}{5.03798pt}{1.05461pt}{6.82881pt}{-1.15456pt}{6.82881pt}\pgfsys@lineto{-5.69089pt}{6.82881pt}\pgfsys@stroke\pgfsys@invoke{ }{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{}{}}{{}{}}{}{}{{}}{{}}{{}{}}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{ } {\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ } {{}} {{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-2.95834pt}{6.82881pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope 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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.53043pt}{-17.49919pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{{\scriptstyle k+1-i}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,. Then, $X(k;i)=0$ for $1<i<k+1$ and

[TABLE]

Proof.

$A_{2}$ webs appearing in the RHS of Lemma 3.9 contains $t_{{-}}^{{+}{+}}$ if $j\geq 1$ in the top side. Thus,

[TABLE]

by applying Lemma 3.9 to $P_{{+}^{k}}^{{+}^{k}}$. The $A_{2}$ web in the second term has $X(k-1;i)$. We obtain proof by induction on $k$. ∎

Proposition 3.11** (Kim [Kim07, Theorem 3.3]).**

For any positive integers $k$ and $l$,

[TABLE]

Proof.

We decompose $P_{{+}^{k+1}{-}^{l}}^{{+}^{k+1}{-}^{l}}$ into $(\mathbf{1}_{+}\otimes P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})P_{{+}^{k+1}{-}^{l}}^{{+}^{k+1}{-}^{l}}(\mathbf{1}_{+}\otimes P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})$ and expand the middle $A_{2}$ clasp $P_{{+}^{k+1}{-}^{l}}^{{+}^{k+1}{-}^{l}}$ by using Definition 2.7. We calculate the following $A_{2}$ web:

[TABLE]

By Lemma 3.10, $X(k,l;i)=0$ for $i>1$ and

[TABLE]

We can easily calculate $X(k,l;0)$ from Definition 2.2:

[TABLE]

We complete the proof by substituting the above solutions of $X(k,l;i)$ into

[TABLE]

∎

Lemma 3.12**.**

[TABLE]

Proof.

Apply Proposition 3.11 to $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ as follows:

[TABLE]

Once again, we apply Proposition 3.11 to $P_{{-}^{l}{+}^{k-1}}^{{-}^{l}{+}^{k-1}}$:

[TABLE]

∎

Lemma 3.13** (Otsuki-Yamada [OY97, Lemma 5.3]).**

[TABLE]

Proof.

The $A_{2}$ web in the left-hand side should be expressed as a scalar multiplication of $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$. This constant can be calculated by taking the closure of the $A_{2}$ webs in the both sides. We remark that the value of the closure of $P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}}$ is $\frac{[k+1][l+1][k+l+2]}{[2]}$. ∎

Proposition 3.14**.**

For any positive integers $k,l\geq 1$,

[TABLE]

Proof.

By Proposition 3.11,

[TABLE]

It is easy to see that the left-hand side is an idempotent and $X\langle P_{(k,l)}\rangle-\langle P_{(k+1,l)}\rangle$ in $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$. We denote the right-hand side of the above $A_{2}$ web by $g$. We consider morphisms $p_{1}\colon({+}^{k+1}{-}^{l},g)\to P_{(k-1,l+1)}$ and $p_{2}\colon({+}^{k+1}{-}^{l},g)\to P_{(k,l-1)}$ defined by

[TABLE]

Let us confirm that $p_{1}$ and $p_{2}$ are morphisms in $\operatorname{Kar}(\mathbf{Sp}_{q})$. By a similar way to (3.8) and using Lemma 3.12,

[TABLE]

By Lemma 3.12 and Lemma 3.13,

[TABLE]

Let us define $\iota_{1}\colon P_{(k-1,l+1)}\to({+}^{k+1}{-}^{l},g)$ and $\iota_{2}\colon P_{(k,l-1)}\to({+}^{k+1}{-}^{l},g)$ by $(\mathbf{1}_{+}\otimes P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})(t_{{-}}^{{+}{+}}\otimes P_{{+}^{k-1}{-}^{l}}^{{+}^{k-1}{-}^{l}})P_{{+}^{k-1}{-}^{l+1}}^{{+}^{k-1}{-}^{l+1}}$ and $(\mathbf{1}_{+}\otimes P_{{+}^{k}{-}^{l}}^{{+}^{k}{-}^{l}})(b^{{+}{-}}\otimes P_{{+}^{k}{-}^{l-1}}^{{+}^{k}{-}^{l-1}})$, respectively. Because these webs are obtained by turning the webs in $p_{1}$ and $p_{2}$ upside down, we can confirm these maps are morphisms in $\operatorname{Kar}(\mathbf{Sp}_{q})$ by the same calculation in the above. The rest of the proof is to confirm that $p_{1}$, $p_{2}$, $\iota_{1}$, and $\iota_{2}$ satisfy the condition in Lemma 3.5. It is easy to see that $p_{i}\circ\iota_{j}=0$ if $i\neq j$. $p_{1}\circ\iota_{1}=P_{{+}^{k-1}{-}^{l+1}}^{{+}^{k-1}{-}^{l+1}}$ and $p_{2}\circ\iota_{2}=P_{{+}^{k}{-}^{l-1}}^{{+}^{k}{-}^{l-1}}$ are derived from (3.8) and Lemma 3.13, respectively. By Proposition 3.11,

[TABLE]

The last equation uses a formula $[a][b]=\sum_{i=1}^{a}[a+b-(2i-1)]$ where $a$ and $b$ are integers. Thus, the condition in Lemma 3.5 is satisfied and we obtained the isomorphism $({+}^{k+1}{-}^{l},g)\cong P_{(k-1,l+1)}\oplus P_{(k,l-1)}$ in $\operatorname{Kar}(\mathbf{Sp}_{q})$. In terms of $K_{0}(\operatorname{Kar}(\mathbf{Sp}_{q}))$, the equation (3.9) is interpreted as

[TABLE]

∎