Combinatorial Representation Theory

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

Last update: 17 September 2013

Appendix A

A3. The flag variety, unipotent varieties, and Springer theory for $GL(n,ℂ)$

Borel subgroups, Cartan subgroups, and unipotent elements

The groups $$\begin{array}{ccc}{B}_n& =& \left\{\left(\begin{array}{cccc}*& *& \cdots & *\\ 0& *& & ⋮\\ ⋮& & \ddots & *\\ 0& \cdots & 0& *\end{array}\right)\right\},\\ T_n& =& \left\{\left(\begin{array}{cccc}*& 0& \cdots & 0\\ 0& *& & ⋮\\ ⋮& & \ddots & 0\\ 0& \cdots & 0& *\end{array}\right)\right\},\\ U_n& =& \left\{\left(\begin{array}{cccc}1& *& \cdots & *\\ 0& 1& & ⋮\\ ⋮& & \ddots & *\\ 0& \cdots & 0& 1\end{array}\right)\right\},\end{array}$$ are the subgroups of $GL(n,ℂ)$ consisting of upper triangular, diagonal, and upper unitriangular matrices, respectively.

A Borel subgroup of $GL(n,ℂ)$ is a subgroup which is conjugate to $B_n\text{.}$

A Cartan subgroup of $GL(n,ℂ)$ is a subgroup which is conjugate to $T_n\text{.}$

A matrix $u\in GL(n,ℂ)$ is unipotent if it is conjugate to an upper unitriangular matrix.

The flag variety

There is a one-to-one correspondence between each of the following sets:

(1)	$ℬ=\left\{\text{Borel subgroups of}\hspace{0.17em}GL(n,ℂ)\right\},$
(2)	$G/B,$ where $G=GL(n,ℂ)$ and $B=B_n,$
(3)	$\{\text{flags}\hspace{0.17em}0\subseteq V_1\subseteq V_2\subseteq \cdots \subseteq V_n={ℂ}^n\hspace{0.17em}\text{such that dim}\left(V_i\right)=i\}\text{.}$

Each of these sets naturally has the structure of a complex algebraic variety, which is called the flag variety.

The unipotent varieties

Given a unipotent element $u\in GL(n,ℂ)$ with Jordan blocks given by the partition $\mu =({\mu }_1,\dots ,{\mu }_{\ell })$ of $n,$ define an algebraic variety $${ℬ}_{\mu }={ℬ}_u=\left\{\text{Borel subgroups of}\hspace{0.17em}GL(n,ℂ)\hspace{0.17em}\text{which contain}\hspace{0.17em}u\right\}\text{.}$$ By conjugation, the structure of the subvariety ${ℬ}_u$ of the flag variety depends only on the partition $\mu \text{.}$ Thus ${ℬ}_{\mu }$ is well defined, as an algebraic variety.

Springer theory

It is a deep theorem of Springer [Spr1978] (which holds in the generality of semisimple algebraic groups and their corresponding Weyl groups) that there is an action of the symmetric group $S_n$ on the cohomology $H^{*}\left({ℬ}_u\right)$ of the variety ${ℬ}_u\text{.}$ This action can be interpreted nicely as follows. The imbedding $${ℬ}_u\subseteq ℬ\text{induces a surjective map}{H}^{*}\left(ℬ\right)⟶H^{*}\left({ℬ}_u\right)\text{.}$$ It is a famous theorem of Borel that there is a ring isomorphism $$\begin{array}{cc}{H}^{*}\left(ℬ\right)\cong ℂ[x_1,\dots ,x_n]/I^{+},& \text{(}A\text{3.1)}\end{array}$$ where $I^{+}$ is the ideal generated by symmetric functions without constant term. It follows that $H^{*}\left({ℬ}_u\right)$ is also a quotient of $ℂ[x_1,\dots ,x_n]\text{.}$ From the work of Kraft [Kra1980], DeConcini and Procesi [DPr1981] and Tanisaki [Tan1982], one has that the ideal ${𝒯}_u$ which it is necessary to quotient by in order to obtain an isomorphism $$H^{*}\left(ℬ\right)\cong ℂ[x_1,\dots ,x_n]/{𝒯}_u,$$ can be described explicitly.

The symmetric group $S_n$ acts on the polynomial ring $ℂ[x_1,\dots ,x_n]$ by permuting the variables. It turns out that the ideal ${𝒯}_u$ remains invariant under this action, thus yielding a well defined action of $S_n$ on $ℂ[x_1,\dots ,x_n]/{𝒯}_u\text{.}$ This action coincides with the Springer action on $H^{*}\left({ℬ}_u\right)\text{.}$ Hotta and Springer [HSp1977] have established that, if $u$ is a unipotent element of shape $\mu$ then, for every permutation $w\in S_n,$ $$\sum _i{q}^i\epsilon \left(w\right)\text{trace}(w^{-1},H^{2i}\left({ℬ}_u\right))=\sum _{\lambda ⊢n}{\stackrel{\sim }{K}}_{\lambda \mu }\left(q\right){\chi }^{\lambda }\left(w\right),$$ where

$\epsilon \left(w\right)$ is the sign of the permutation $w,$ $\text{trace}(w^{-1},H^{2i}\left({ℬ}_u\right))$ is the trace of the action of $w^{-1}$ on $H^{2i}\left({ℬ}_u\right),$ ${\chi }^{\lambda }\left(w\right)$ is the irreducible character of the symmetric group evaluated at $w,$ and ${\stackrel{\sim }{K}}_{\lambda \mu }\left(q\right)$ is a variant of the Kostka-Foulkes polynomial, see [Mac1995] III §7 Ex. 8, and §6.

It follows from this discussion and some basic facts about the polynomials ${\stackrel{\sim }{K}}_{\lambda \mu }\left(q\right)$ that the top degree cohomology group in $H^{*}\left({ℬ}_{\mu }\right)$ is a realization of the irreducible representation of $S_n$ indexed by $\mu ,$ $$S^{\mu }\cong H^{\text{top}}\left({ℬ}_{\mu }\right)\text{.}$$ This construction of the irreducible modules of $S_n$ is the Springer construction.

References

See [Mac1995] II §3 Ex. 1 for a description of the variety ${ℬ}_u$ and its structure. The theorem of Borel stated in (A3.1) is given in [Bor0051508] and [BGG1973]. The references quoted in the text above will provide a good introduction to the Springer theory. The beautiful combinatorics of Springer theory has been studied by Barcelo [Bar1993], Garsia-Procesi [GPr1992], Lascoux [Las1989], Lusztig [LSp1983], Shoji [Sho1979], Spaltenstein [Spa1976], Weyman [Wey1989], and others.

Notes and references

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..

page history