Last update: 17 September 2013
A $k\text{-braid}$ is viewed as two rows of $k$ vertices, one above the other, and $k$ strands that connect top vertices to bottom vertices in such a way that each vertex is incident to precisely one strand. Strands cross over and under each other in three-space as they pass from one vertex to the next. $$\begin{array}{ccccc}{t}_{1}=& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n& ,\phantom{\rule{2em}{0ex}}{t}_{2}=& \n\n\n\n\n\n\n\n\n\n\n\n& \text{.}\end{array}$$ We multiply $k\text{-braids}$ ${t}_{1}$ and ${t}_{2}$ using the concatenation product given by identifying the vertices in the top row of ${t}_{2}$ with the corresponding vertices in the bottom row of ${t}_{1}$ to obtain the product ${t}_{1}{t}_{2}\text{.}$ $$\begin{array}{ccc}{t}_{1}{t}_{2}=& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n& ,\end{array}$$
Given a permutation $w\in {S}_{k}$ we will make a $k\text{-braid}$ ${T}_{w}$ by tracing the edges in order from left to right across the top row. Any time an edge that we are tracing crosses an edge that has been already traced we raise the pen briefly so that the edge being traced goes under the edge which is already there. Applying this process to all of the permutations in ${S}_{k}$ produces a set of $k!$ braids. $$\begin{array}{cccc}w=& \n\n\n\n\n\n\n& \phantom{\rule{2em}{0ex}}{T}_{2}=& \n\n\n\n\n\n\n\n\n\n\n\n\n\end{array}$$
Fix $q\in \u2102\text{.}$ The Iwahori-Hecke algebra ${H}_{k}\left(q\right)$ of type ${A}_{k-1}$ is the span of the $k!$ braids produced by tracing permutations in ${S}_{k}$ with multiplication determined by the braid multiplication and the following identity. $$\begin{array}{cccccc}\n\n\n\n\n\n\n& =(q-1)& \n\n\n\n\n& +q& \n\n\n\n\n\n\n& \text{.}\end{array}$$ This identity can be applied in any local portion of the braid.
Theorem B5.1. The algebra ${H}_{k}\left(q\right)$ is the associative algebra over $\mathbb{Z}$ presented by generators ${T}_{1},\dots ,{T}_{k-1}$ and relations $$\begin{array}{cc}{T}_{i}{T}_{j}={T}_{j}{T}_{i},& \text{for}\hspace{0.17em}|i-j|>1,\\ {T}_{i}{T}_{i+1}{T}_{i}={T}_{i+1}{T}_{i}{T}_{i+1},& \text{for}\hspace{0.17em}1\le i\le n-2,\\ {T}_{i}^{2}=(q-1){T}_{i}+q,& \text{for}\hspace{0.17em}2\le i\le n\text{.}\end{array}$$
The Iwahori-Hecke algebra of type $A$ is a $q\text{-analogue}$ of the group algebra of the symmetric group. If we allow ourselves to be imprecise (about the limit) we can write $$\underset{q\to 1}{\text{lim}}{H}_{k}\left(q\right)=\u2102{S}_{k}\text{.}$$
Let $q$ be a power of a prime $G=GL(n,{\mathbb{F}}_{q})$ where Fq is the finite field with $q$ elements. Let $B$ be the subgroup of upper triangular matrices in $G$ and let ${1}_{B}^{G}$ be the trivial representation of $B$ induced to $G,$ i.e. the $G\text{-module}$ given by $${1}_{B}^{G}=\u2102\text{-span}\left\{gB\hspace{0.17em}\right|\hspace{0.17em}g\in G\},$$ where $G$ acts on the cosets by left multiplication. Using the description, see §B3, of ${H}_{n}\left(q\right)$ as a double coset algebra one gets an action of ${H}_{n}\left(q\right)$ on ${1}_{B}^{G},$ by right multiplication. This action commutes with the $G$ action.
Theorem B5.2.
(a) | The action of ${H}_{n}\left(q\right)$ on ${1}_{B}^{G}$ generates ${\text{End}}_{G}\left({1}_{B}^{G}\right)\text{.}$ |
(b) | The action of $G$ on ${1}_{B}^{G}$ generates ${\text{End}}_{{H}_{n}\left(q\right)}\left({1}_{B}^{G}\right)\text{.}$ |
This theorem gives a “duality” between $GL(n,{\mathbb{F}}_{q})$ and ${H}_{n}\left(q\right)$ which is similar to a Schur-Weyl duality, but it differs in a crucial way: the representation ${1}_{B}^{G}$ is not a tensor power representation, and thus this is not yet realizing ${H}_{n}\left(q\right)$ as a tensor power centralizer.
The following result gives a true analogue of the Schur-Weyl duality for the Iwahori-Hecke algebra of type $A,$ it realizes ${H}_{k}\left(q\right)$ as a tensor power centralizer. Assume that $q\in \u2102$ is not $0$ and is not a root of unity. Let ${U}_{q}{\U0001d530\U0001d529}_{n}$ be the Drinfel’d-Jimbo quantum group of type ${A}_{n-1}$ and let $V$ be the $n\text{-dimensional}$ irreducible representation of ${U}_{q}{\U0001d530\U0001d529}_{n}$ with highest weight ${\omega}_{1}\text{.}$ There is an action, see [CPr1994], of ${H}_{k}\left({q}^{2}\right)$ on ${V}^{\otimes k}$ which commutes with the ${U}_{q}{\U0001d530\U0001d529}_{n}$ action.
Theorem B5.3.
(a) | The action of ${H}_{k}\left({q}^{2}\right)$ on ${V}^{\otimes k}$ generates ${\text{End}}_{{U}_{q}{\U0001d530\U0001d529}_{n}}\left({V}^{\otimes k}\right)\text{.}$ |
(b) | The action of ${U}_{q}{\U0001d530\U0001d529}_{n}$ on ${V}^{\otimes k}$ generates ${\text{End}}_{{H}_{k}\left({q}^{2}\right)}\left({V}^{\otimes k}\right)\text{.}$ |
Theorem B5.4. The Iwahori-Hecke algebra of type ${A}_{k-1},$ ${H}_{k}\left(q\right),$ is semisimple if and only if $q\ne 0$ and $q$ is not a $j\text{th}$ root of unity for any $2\le j\le n\text{.}$
Partial results for ${H}_{k}\left(q\right)$
The following results giving answers to the main questions (Ia-c) for the Iwahori-Hecke algebras of type $A$ hold when $q$ is such that ${H}_{k}\left(q\right)$ is semisimple.
(a) | How do we index/count them? |
There is a bijection $$\text{Partitions}\hspace{0.17em}\lambda \hspace{0.17em}\text{of}\hspace{0.17em}n\phantom{\rule{2em}{0ex}}\stackrel{1-1}{\u27f7}\phantom{\rule{2em}{0ex}}\text{Irreducible representations}\hspace{0.17em}{H}^{\lambda}$$ | |
(b) | What are their dimensions? |
The dimension of the irreducible representation ${H}^{\lambda}$ is given by $$\begin{array}{ccc}\text{dim}\left({H}^{\lambda}\right)& =& \text{\# of standard tableaux of shape}\hspace{0.17em}\lambda \\ & =& \frac{n!}{{\prod}_{x\in \lambda}{h}_{x}},\end{array}$$ | |
(c) | What are their characters? |
For each partition $\mu =({\mu}_{1},{\mu}_{2},\dots ,{\mu}_{\ell})$ of $k$ let ${\chi}^{\lambda}\left(\mu \right)$ be the character of the irreducible representation ${H}^{\lambda}$ evaluated at the element ${T}_{{\gamma}_{\mu}}$ where ${\gamma}_{\mu}$ is the permutation $${\gamma}_{\mu}=\underset{\underset{{\mu}_{1}}{\u23df}}{\begin{array}{c}\n\n\n\n\n\n\end{array}}\phantom{\rule{1em}{0ex}}\underset{\underset{{\mu}_{2}}{\u23df}}{\begin{array}{c}\n\n\n\n\n\end{array}}\phantom{\rule{1em}{0ex}}\cdots \phantom{\rule{1em}{0ex}}\underset{\underset{{\mu}_{\ell}}{\u23df}}{\begin{array}{c}\n\n\n\end{array}}\text{.}$$ |
References
The book [CPr1994] contains a treatment of the Schur-Weyl duality type theorem given above. See also the references there. Several basic results on the Iwahori-Hecke algebra are given in the book [GHJ1989]. The theorem giving the explicit values of $q$ such that ${H}_{k}\left(q\right)$ is semisimple is due to Gyoja and Uno [GUn1989]. The character formula given above is due to Roichman [Roi1997]. See [Ram1997-2] for an elementary proof.
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..