Searcharxiv⌕ Search

arXiv subjects

Anthony Henderson

Publications and source records attributed to Anthony Henderson.

30 records · Page 2Linked to original sources
arXiv2010

Pieces of nilpotent cones for classical groups

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.

math.RT↗
arXiv2010

Exterior powers of the reflection representation in the cohomology of Springer fibres

Let $H^*(\calB_e)$ be the cohomology of the Springer fibre for the nilpotent element $e$ in a simple Lie algebra $\g$, on which the Weyl group $W$ acts by the Springer representation. Let $Λ^i V$ denote the $i$th exterior power of the reflection representation of $W$. We determine the degrees in which $Λ^i V$ occurs in the graded representation $H^*(\calB_e)$, under the assumption that $e$ is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if $\g$ is of type A, B, or C. This partially verifies a conjecture of Lehrer--Shoji, and extends the results of Solomon in the $e=0$ case and Lehrer--Shoji in the $i=1$ case. The proof proceeds by showing that $(H^*(\calB_e) \otimes Λ^* V)^W$ is a free exterior algebra on its subspace $(H^*(\calB_e)\otimes V)^W$.

math.RT↗
arXiv2008

The equivariant Euler characteristic of real Coxeter toric varieties

Let $W$ be a Weyl group, and let $\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\CT_W(\R)$ is a smooth, connected, compact manifold with a $W$-action. We give a formula for the equivariant Euler characteristic of $\CT_W(\R)$ as a generalised character of $W$. In type $A_{n-1}$ for $n$ odd, one obtains a generalised character of $\Sym_n$ whose degree is (up to sign) the $n^{\text{th}}$ Euler number.

math.RT↗
arXiv2007

The cohomology of real De Concini-Procesi models of Coxeter type

We study the rational cohomology groups of the real De Concini-Procesi model corresponding to a finite Coxeter group, generalizing the type-A case of the moduli space of stable genus 0 curves with marked points. We compute the Betti numbers in the exceptional types, and give formulae for them in types B and D. We give a generating-function formula for the characters of the representations of a Coxeter group of type B on the rational cohomology groups of the corresponding real De Concini-Procesi model, and deduce the multiplicities of one-dimensional characters in the representations, and a formula for the Euler character. We also give a moduli space interpretation of this type-B variety, and hence show that the action of the Coxeter group extends to a slightly larger group.

math.RT↗
arXiv2006

Bases for certain cohomology representations of the symmetric group

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.

math.RT↗
arXiv2005

Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of type A

The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.

math.RT↗
arXiv2003

Representations of wreath products on cohomology of De Concini-Procesi compactifications

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.

math.RT↗
arXiv2002

Two-row nilpotent orbits of cyclic quivers

We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of $GL_d$ corresponding to nilpotents with at most two Jordan blocks are multiplicity-free.

math.RT↗
arXiv2002

K^F-invariants in irreducible representations of G^F, when G=GL_n

Using a general result of Lusztig, we give explicit formulas for the dimensions of K^F-invariants in irreducible representations of G^F, when G=GL_n, F:G->G is a Frobenius map, and K is an F-stable subgroup of finite index in G^theta for some involution theta:G->G commuting with F. The proofs use some combinatorial facts about characters of symmetric groups.

math.RT↗