## Representations of the symmetric group

Last update: 8 October 2012

## Representations of the symmetric group

The symmetric group is ${S}_{d}=\text{Aut}\phantom{\rule{0.2em}{0ex}}\left(\left\{1,2,\dots ,d\right\}\right)\text{.}$

The symmetric group is presented by generators

$si= 1 ˙ ˙ ˙ ˙ i i+1 ˙ ˙ ˙ d , i=1,2,…,d-1$

with relations

$(✶) si2=1andsi sj=sjsiif j≠i±1,si si+1si= si+1sisi+1.$

The degenerate affine Hecke algebra ${ℋ}_{d}$ 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 ${ℋ}_{d}$ with additional relation

$x1=0$

i.e.

$ℋd⟶ℂSd si⟼si x1⟼0 xj ⟼ ∑1≤i

### KLR Quiver Hecke algebras

 $I=vertex set of Dynkin diagram= {colours}.$

The KLR quiver Hecke algebra ${R}_{d}$ is given by generators

$y1,…,yd, euforu∈Id, ψ1,…,ψd-1$

with relations

$yiyj=yjyi, euev=δuv eu,1=∑u∈Id eu, euyi=yieu, euψr=ψresru, ψryi=yiψr ifi≠r,r+1, ψryreu= { (yr+1ψr+1)eu , if(ur,ur+1) =(ur,ur) , yr+1ψreu , otherwise. ψryr+1eu= { (yrψr-1)eu , if(ur,ur+1) =(ur,ur) , yrψreu , otherwise. ψrψs=ψsψr, ifs≠r,r±1, ψr2eu= { 0, if(ur,ur+1) =(ur,ur) , (yr+1-yr)eu , if(ur,ur+1) is ur ur+1 , -(yr+1-yr)eu , if(ur,ur+1) is ur ur+1 , eu , otherwise. ψrψr+1ψr eu= { ( ψr+1 ψr ψr+1 +1 ) eu, if ( ur,ur+1, ur+2 ) = ( ur,ur+1, ur ) with ur ur+1 , ( ψr+1 ψr ψr+1 -1 ) eu, if ( ur,ur+1, ur+2 ) = ( ur,ur+1, ur ) with ur ur+1 , ψr+1 ψr ψr+1 eu , otherwise.$

where

$Id= { u=(u1,…,ud) sequences of lengthdinI } sruisuwith urand ur+1switched.$

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

$ℋ^d≅R^d$

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

$ℂ{S}_{d}$ is ${R}_{d}$ for $\begin{array}{cc}Q=& -1 0 1 2 3 \end{array}$ with

$eu=0ifu1≠0 andy1eu=0ifu1=0.$

$Board Beads$

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

$if then or$

A standard tableau of shape $\lambda$ is a runner sequence $T=\left({T}_{1},\dots ,{T}_{d}\right)$ which results in $\lambda \text{.}$

Define

$Rdλ=span { vT∣T is a standard of shapeλ }$

with

$euvT= δuTvT, yivT=0, ψrvT= { vsrT, ifsrT is a standard shape, 0, otherwise.$

(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 $ℂ{S}_{d}$-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.