Representations of the symmetric group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 8 October 2012

Representations of the symmetric group

The symmetric group is $S_d=\text{Aut}\left(\{1,2,\dots ,d\}\right)\text{.}$

The symmetric group is presented by generators

$$\begin{array}{ccc}{s}_i=& 1 ˙ ˙ ˙ ˙ i i+1 ˙ ˙ ˙ d ,& i=1,2,\dots ,d-1\end{array}$$

with relations

$$\begin{array}{cc}\text{(}✶\text{)}& s_i^2=1\text{and}{s}_i{s}_j=s_j{s}_i\text{if}j\ne i\pm 1,s_i{s}_{i+1}{s}_i=s_{i+1}{s}_i{s}_{i+1}\text{.}\end{array}$$

The degenerate affine Hecke algebra ${\mathscr{H}}_d$ is given by generators $x_1,\dots ,x_d$ and $s_1,\dots ,s_{d-1}$ with relations $x_i{x}_j=x_j{x}_i$ and $\left(✶\right)$ and $s_i{x}_j=x_j{s}_i$ if $j\ne i,$ $i+1,$ and $s_i{x}_i=x_{i+1}{s}_i+1,$ $s_i{x}_{i+1}=x_i{s}_i-1\text{.}$

The group algebra of $S_d$ is ${\mathscr{H}}_d$ with additional relation

$$x_1=0$$

i.e.

$$\begin{array}{ccc}{\mathscr{H}}_d& ⟶& ℂS_d\\ s_i& ⟼& s_i\\ x_1& ⟼& 0\\ x_j& ⟼& \sum _{1\le i<j}{s}_{ij}& \text{where}{s}_{ij}=& 1 ˙ ˙ i ˙ ˙ ˙ ˙ ˙ j ˙ ˙ d \end{array}$$

KLR Quiver Hecke algebras

$$$$ $$I=\text{vertex set of Dynkin diagram}=\left\{\text{colours}\right\}\text{.}$$

The KLR quiver Hecke algebra $R_d$ is given by generators

$$y_1,\dots ,y_d,e_u\text{for}u\in I^d,{\psi }_1,\dots ,{\psi }_{d-1}$$

with relations

$$\begin{array}{c}{y}_i{y}_j=y_j{y}_i,e_u{e}_v={\delta }_{uv}{e}_u,1=\sum _{u\in I^d}{e}_u,\\ e_u{y}_i=y_i{e}_u,e_u{\psi }_r={\psi }_r{e}_{s_{r}u},{\psi }_r{y}_i=y_i{\psi }_r\text{if}i\ne r,r+1,\\ {\psi }_r{y}_r{e}_u=\{\begin{array}{cc}(y_{r+1}{\psi }_r+1)e_u,& \text{if}(u_r,u_{r+1})=(u_r,u_r),\\ y_{r+1}{\psi }_r{e}_u,& \text{otherwise.}\end{array}\\ {\psi }_r{y}_{r+1}{e}_u=\{\begin{array}{cc}(y_r{\psi }_r-1)e_u,& \text{if}(u_r,u_{r+1})=(u_r,u_r),\\ y_r{\psi }_r{e}_u,& \text{otherwise.}\end{array}\\ {\psi }_r{\psi }_s={\psi }_s{\psi }_r,\text{if}s\ne r,r\pm 1,\\ {\psi }_r^2{e}_u=\{\begin{array}{cc}0,& \text{if}(u_r,u_{r+1})=(u_r,u_r),\\ (y_{r+1}-y_r)e_u,& \text{if}(u_r,u_{r+1})\text{is} ur ur+1 ,\\ -(y_{r+1}-y_r)e_u,& \text{if}(u_r,u_{r+1})\text{is} ur ur+1 ,\\ e_u,& \text{otherwise.}\end{array}\\ {\psi }_r{\psi }_{r+1}{\psi }_r{e}_u=\{\begin{array}{cc}({\psi }_{r+1}{\psi }_r{\psi }_{r+1}+1)e_u,& \text{if}(u_r,u_{r+1},u_{r+2})=(u_r,u_{r+1},u_r)\text{with} ur ur+1 ,\\ ({\psi }_{r+1}{\psi }_r{\psi }_{r+1}-1)e_u,& \text{if}(u_r,u_{r+1},u_{r+2})=(u_r,u_{r+1},u_r)\text{with} ur ur+1 ,\\ {\psi }_{r+1}{\psi }_r{\psi }_{r+1}{e}_u,& \text{otherwise.}\end{array}\end{array}$$

where

$$\begin{array}{c}{I}^d=\{u=(u_1,\dots ,u_d)\text{sequences of length}d\text{in}I\}\\ s_{r}u\text{is}u\text{with}{u}_r\text{and}{u}_{r+1}\text{switched.}\end{array}$$

If $\begin{array}{cc}Q=& 4 3 0 1 2 \end{array}$ then, after a completion,

$${\hat{\mathscr{H}}}_d\cong {\hat{R}}_d$$

$\text{(}ℂ\left[[x_1,\dots ,x_d]\right]$ is a completion of $ℂ[x_1,\dots ,x_d]$ which contains $\frac{1}{1-x_1}=1+x_1+x_1^2+\dots \text{)}$

$ℂS_d$ is $R_d$ for $\begin{array}{cc}Q=& -1 0 1 2 3 \end{array}$ with

$$e_u=0\text{if}{u}_1\ne 0\text{and}{y}_1{e}_u=0\text{if}{u}_1=0\text{.}$$

The Glass Bead game

$$\begin{array}{cc} & \\ \text{Board}& \text{Beads}\end{array}$$

A skew shape is a configuration of beads $\lambda$ such that any two beads on the same runner are separated by two beads

$$\begin{array}{cccccc}\text{if}& & \text{then}& & \text{or}& \end{array}$$

A standard tableau of shape $\lambda$ is a runner sequence $T=(T_1,\dots ,T_d)$ which results in $\lambda \text{.}$

Define

$$R_d^{\lambda }=\text{span}\{v_T\mid T\text{is a standard of shape}\lambda \}$$

with

$$e_u{v}_T={\delta }_{uT}{v}_T,y_i{v}_T=0,{\psi }_r{v}_T=\{\begin{array}{cc}{v}_{s_{r}T},& \text{if}{s}_{r}T\text{is a standard shape,}\\ 0,& \text{otherwise.}\end{array}$$

(Kleshehev-Ram) $R_d^{\lambda }$ are simple $R_d$-modules.

Young's lattice

$$-2 -1 0 1 2 3$$ $$∅ 0 -1 1 -2 2 -1 1 -3 3 -1 1 -2 2 0$$

Standard tableaux of shape $\lambda$ correspond to paths from 0 to $\lambda \text{.}$

$R_d^{\lambda }$ for $\lambda$ in Young's Lattice (with $d$ beads) are all simple $ℂS_d$-modules.

Notes and References

These are a typed version of lecture notes for the second in a series of lectures given at the Brazil Algebra Conference, Salvador, 17 July 2012.

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