Last update: 17 September 2013

*Borel subgroups, Cartan subgroups, and unipotent elements*

The groups $$\begin{array}{ccc}{B}_{n}& =& \left\{\left(\begin{array}{cccc}*& *& \cdots & *\\ 0& *& & \vdots \\ \vdots & & \ddots & *\\ 0& \cdots & 0& *\end{array}\right)\right\},\\ {T}_{n}& =& \left\{\left(\begin{array}{cccc}*& 0& \cdots & 0\\ 0& *& & \vdots \\ \vdots & & \ddots & 0\\ 0& \cdots & 0& *\end{array}\right)\right\},\\ {U}_{n}& =& \left\{\left(\begin{array}{cccc}1& *& \cdots & *\\ 0& 1& & \vdots \\ \vdots & & \ddots & *\\ 0& \cdots & 0& 1\end{array}\right)\right\},\end{array}$$ are the subgroups of $GL(n,\u2102)$ consisting of upper triangular, diagonal, and upper unitriangular matrices, respectively.

A *Borel subgroup* of $GL(n,\u2102)$ is a subgroup which
is conjugate to ${B}_{n}\text{.}$

A *Cartan subgroup* of $GL(n,\u2102)$ is a subgroup which is conjugate to
${T}_{n}\text{.}$

A matrix $u\in GL(n,\u2102)$
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) | $\mathcal{B}=\left\{\text{Borel subgroups of}\hspace{0.17em}GL(n,\u2102)\right\},$ |

(2) | $G/B,$ where $G=GL(n,\u2102)$ and $B={B}_{n},$ |

(3) | $\{\text{flags}\hspace{0.17em}0\subseteq {V}_{1}\subseteq {V}_{2}\subseteq \cdots \subseteq {V}_{n}={\u2102}^{n}\hspace{0.17em}\text{such that dim}\left({V}_{i}\right)=i\}\text{.}$ |

*The unipotent varieties*

Given a unipotent element $u\in GL(n,\u2102)$ with Jordan blocks given by the partition $\mu =({\mu}_{1},\dots ,{\mu}_{\ell})$ of $n,$ define an algebraic variety $${\mathcal{B}}_{\mu}={\mathcal{B}}_{u}=\left\{\text{Borel subgroups of}\hspace{0.17em}GL(n,\u2102)\hspace{0.17em}\text{which contain}\hspace{0.17em}u\right\}\text{.}$$ By conjugation, the structure of the subvariety ${\mathcal{B}}_{u}$ of the flag variety depends only on the partition $\mu \text{.}$ Thus ${\mathcal{B}}_{\mu}$ 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 ${H}^{*}\left({\mathcal{B}}_{u}\right)$ of the variety ${\mathcal{B}}_{u}\text{.}$ This action can be interpreted nicely as follows. The imbedding $${\mathcal{B}}_{u}\subseteq \mathcal{B}\phantom{\rule{1em}{0ex}}\text{induces a surjective map}\phantom{\rule{1em}{0ex}}{H}^{*}\left(\mathcal{B}\right)\u27f6{H}^{*}\left({\mathcal{B}}_{u}\right)\text{.}$$ It is a famous theorem of Borel that there is a ring isomorphism $$\begin{array}{cc}{H}^{*}\left(\mathcal{B}\right)\cong \u2102[{x}_{1},\dots ,{x}_{n}]/{I}^{+},& \text{(}A\text{3.1)}\end{array}$$ where ${I}^{+}$ is the ideal generated by symmetric functions without constant term. It follows that ${H}^{*}\left({\mathcal{B}}_{u}\right)$ is also a quotient of $\u2102[{x}_{1},\dots ,{x}_{n}]\text{.}$ From the work of Kraft [Kra1980], DeConcini and Procesi [DPr1981] and Tanisaki [Tan1982], one has that the ideal ${\mathcal{T}}_{u}$ which it is necessary to quotient by in order to obtain an isomorphism $${H}^{*}\left(\mathcal{B}\right)\cong \u2102[{x}_{1},\dots ,{x}_{n}]/{\mathcal{T}}_{u},$$ can be described explicitly.

The symmetric group ${S}_{n}$ acts on the polynomial ring $\u2102[{x}_{1},\dots ,{x}_{n}]$ by permuting the variables. It turns out that the ideal ${\mathcal{T}}_{u}$ remains invariant under this action, thus yielding a well defined action of ${S}_{n}$ on $\u2102[{x}_{1},\dots ,{x}_{n}]/{\mathcal{T}}_{u}\text{.}$ This action coincides with the Springer action on ${H}^{*}\left({\mathcal{B}}_{u}\right)\text{.}$ Hotta and Springer [HSp1977] have established that, if $u$ is a unipotent element of shape $\mu $ then, for every permutation $w\in {S}_{n},$ $$\sum _{i}{q}^{i}\epsilon \left(w\right)\text{trace}({w}^{-1},{H}^{2i}\left({\mathcal{B}}_{u}\right))=\sum _{\lambda \u22a2n}{\stackrel{\sim}{K}}_{\lambda \mu}\left(q\right){\chi}^{\lambda}\left(w\right),$$ where

$\epsilon \left(w\right)$ is the sign of the permutation $w,$
$\text{trace}({w}^{-1},{H}^{2i}\left({\mathcal{B}}_{u}\right))$ is the trace of the action of ${w}^{-1}$ on ${H}^{2i}\left({\mathcal{B}}_{u}\right),$ ${\chi}^{\lambda}\left(w\right)$ 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. |

**References**

See [Mac1995] II §3 Ex. 1 for a description of the variety ${\mathcal{B}}_{u}$ 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.

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..