No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
The symmetric group representation on cohomology of the regular elements of a maximal torus of the special linear group
Abstract
We give a formula for the character of the representation of the symmetric group on each isotypic component of the cohomology of the set of regular elements of a maximal torus of , with respect to the action of the centre. Comment: 14 pages; slightly revised in the light of math.RT/0508162
... Denote its kernel by D α . In our work on toric arrangements (see [2] , [5], [6], [7], [8]) we have shown that the Euler characteristic of the open set A := T − ∪ α∈R + D α equals (−1) ℓ |W |. The only proof we know of this fact is via a combinatorial topological construction of Salvetti [4] [3]. ...
We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots.
... Denote its kernel by D α . In our work on toric arrangements (see [2] , [5], [6], [7], [8]) we have shown that the Euler characteristic of the open set A := T − ∪ α∈R + D α equals (−1) ℓ |W |. The only proof we know of this fact is via a combinatorial topological construction of Salvetti [4] [3]. ...
We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots.
... We then set Proof. Everything follows from identity (15). ...
Motivated by the counting formulas of integral
polytopes, as in Brion and Vergne, and Szenes and Vergne, we start to form
the foundations of a theory for toric arrangements, which
may be considered as the periodic version of the
theory of hyperplane arrangements.
We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees.
Let G be a finite group, n a positive integer, Qn(G) the Dowling lattice of rank n based on G and Wn the wreath product group G wr Sn. It is easily seen that Wn acts as a group of automorphisms of Qn(G). This action lifts to a representation of Wn on each homology group of Qn(G). The character values of these representations are computed. Let σ be an element of Wn. Consider σ as an n × n permutation matrix σ whose nonzero entries have been replaced by elements of G. If C is a cycle of σ, the weight of C is the product of the elements of G which lie in the cycle C. The type of C is the conjugacy class of G containing the weight of C. Let cl,u denote the number of l-cycles of σ of type u. The conjugacy class of σ in Wn depends only on the numbers cl,u. For each i = 0, 1,…, n − 1 let (Qn(G))i be the geometric lattice obtained from Qn(G) by deleting ranks i + 1 through n − 1 (so (Qn(G))n − 1 = Qn(G)). Let βi denote the character of the representation of Wn on the unique non-vanishing reduced homology group of Qn(G))i. For each σϵWn, let Bσ(λ) be the polynomial Bσ(λ) = ∑i = 0n − 1βi(σ)) λn − 1 −i. It is shown that where is the number of solutions hϵG to ht = u. The formula (∗) allows for the explicit computation of the characters βi. Using this information, several facts about the characters are deduced. For example, it is shown that the trivial character appears exactly once in each βi and it is shown that βn − 1 can be realized in a simple way as a sum of induced characters.
In an earlier work, the second author proved a general formula for the equivariant Poincaré polynomial of a linear transformation g which normalises a unitary reflection group G, acting on the cohomology of the corresponding hyperplane complement. This formula involves a certain function (called a Z-function below) on the centraliser CG(g), which was proved to exist only in certain cases, for example, when g is a reflection, or is G-regular, or when the centraliser is cyclic. In this work we prove the existence of Z-functions in full generality. Applications include reduction and product formulae for the equivariant Poincaré polynomials. The method is to study the poset L(CG(g)) of subspaces which are fixed points of elements of CG(g). We show that this poset has Euler characteristic 1, which is the key property required for the definition of a Z-function. The fact about the Euler characteristic in turn follows from the ‘join-atom’ property of L(CG(g)), which asserts that if {X1,...,Xk} is any set of elements of L(CG(g)) which are maximal (set theoretically) then their setwise intersection lies in L(CG(g)). 2000 Mathematical Subject Classification: primary 14R20, 55R80; secondary 20C33, 20G40.
The wreath product W(r,n) of the cyclic group of order r and the symmetric group S_n acts on the corresponding projective hyperplane complement, and on its wonderful compactification as defined by De Concini and Procesi. We give a formula for the characters of the representations of W(r,n) on the cohomology groups of this compactification, extending the result of Ginzburg and Kapranov in the r=1 case. As a corollary, we get a formula for the Betti numbers which generalizes the result of Yuzvinsky in the r=2 case. Our method involves applying to the nested-set stratification a generalization of Joyal's theory of tensor species, which includes a link between polynomial functors and plethysm for general r. We also give a new proof of Lehrer's formula for the representations of W(r,n) on the cohomology groups of the hyperplane complement.
On hyperoctahedral hyperplane complements, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif
- G I Lehrer
G. I. Lehrer, On hyperoctahedral hyperplane complements, in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), vol. 47 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1987, pp. 219– 234.
- I G Macdonald
I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ.
Press, second ed., 1995.
School of Mathematics and Statistics, University of Sydney, NSW
2006, AUSTRALIA
E-mail address: anthonyh@maths.usyd.edu.au