Last update: 17 September 2013
Borel subgroups, Cartan subgroups, and unipotent elements
The groups are the subgroups of consisting of upper triangular, diagonal, and upper unitriangular matrices, respectively.
A Borel subgroup of is a subgroup which is conjugate to
A Cartan subgroup of is a subgroup which is conjugate to
A matrix 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:
The unipotent varieties
Given a unipotent element with Jordan blocks given by the partition of define an algebraic variety By conjugation, the structure of the subvariety of the flag variety depends only on the partition Thus is well defined, as an algebraic variety.
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 on the cohomology of the variety This action can be interpreted nicely as follows. The imbedding It is a famous theorem of Borel that there is a ring isomorphism where is the ideal generated by symmetric functions without constant term. It follows that is also a quotient of From the work of Kraft [Kra1980], DeConcini and Procesi [DPr1981] and Tanisaki [Tan1982], one has that the ideal which it is necessary to quotient by in order to obtain an isomorphism can be described explicitly.
The symmetric group acts on the polynomial ring by permuting the variables. It turns out that the ideal remains invariant under this action, thus yielding a well defined action of on This action coincides with the Springer action on Hotta and Springer [HSp1977] have established that, if is a unipotent element of shape then, for every permutation where
is the sign of the permutation
is the trace of the action of on
is the irreducible character of the symmetric group evaluated at and
is a variant of the Kostka-Foulkes polynomial, see [Mac1995] III §7 Ex. 8, and §6.
See [Mac1995] II §3 Ex. 1 for a description of the variety 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..