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Steven Spallone

Publications and source records attributed to Steven Spallone.

At least 19 recordsLinked to original sources
arXiv2026

Stiefel-Whitney classes for symmetric groups

We prove several results about Stiefel-Whitney Classes (SWCs) $w_k(\pi)$ of representations $\pi$ of $S_n$. First, each SWC is polynomial in the character values of $\pi$ at involutions. Next, for a fixed $k$, the proportion of irreducible $\pi$ for which $w_k(\pi)=0$ approaches $100\%$ as $n \to \infty$. A similar result holds for the top SWCs. We also provide a simple criterion which determines the first nonvanishing SWC for a representation. The first four SWCs are computed explicitly. Finally, we give analogues for alternating groups.

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arXiv2026

Characteristic Classes Of Representations Of Lie Groups

An irreducible representation of a reductive Lie algebra, when restricted to a Cartan subalgebra, decomposes into weights with multiplicity. The first part of this paper outlines a procedure to compute symmetric polynomials (e.g., power sums) of this multiset of weights, as functions of the highest weight. Next, let G be a connected reductive complex algebraic group with maximal torus T. We express the restrictions of the Chern classes of irreducible representations of G to T, as polynomial functions in the highest weight. We do the same for Stiefel-Whitney classes of orthogonal representations.

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arXiv2024

Stiefel-Whitney Classes for Finite Symplectic Groups

Let $q$ be an odd prime power, and $G=\text{Sp}(2n,q)$ the finite symplectic group. We give an expression for the total Stiefel-Whitney Classes (SWCs) for orthogonal representations $\pi$ of $G$, in terms of character values of $\pi$ at elements of order $2$. We give "universal formulas'' for the fourth and eighth SWCs. For $n=2$, we compute the subring of the mod $2$ cohomology generated by the SWCs $w_k(\pi)$.

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arXiv2024

The Divisibility of $\mathrm{GL}(n, q)$ Character Values

Let $q$ be a prime power, and $d$ a positive integer. We study the proportion of irreducible characters of $\mathrm{GL}(n,q)$ whose values evaluated on a fixed matrix $g$ are divisible by $d$. As $n$ approaches infinity, this proportion tends to $1$ when $q$ is coprime to $d$. When $q$ and $d$ are not coprime, and $g=1$, this proportion is bounded above by $\frac{1}{q}$.

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arXiv2024

Stiefel-Whitney Classes for Finite Special Linear Groups of Even Rank

We compute the total Stiefel-Whitney Classes (SWCs) for orthogonal representations of special linear groups $\text{SL}(n,q)$ when $n$ and $q$ are odd. These classes are expressed in terms of character values at diagonal elements of order $2$. We give several consequences, and work out the $4$th SWC explicitly, and the $8$th SWC when the $4$th vanishes.

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arXiv2023

On the Divisibility of Degrees of Representations of Lie Algebras

Let $\mathfrak g$ be a reductive Lie algebra, and $m$ a positive integer. There is a natural density of irreducible representations of $\mathfrak g$, whose degrees are not divisible by $m$. For $\mathfrak g=\mathfrak{gl}_n$, this density decays exponentially to $0$ as $n \to \infty$. Similar results hold for simple Lie algebras and Lie groups, and there are versions for self-dual and orthogonal representations.

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arXiv2023

Central Extensions and Cohomology

Let G be a group which is topologically a CW-complex, BG a classifying space for G, and A a discrete abelian group. To a central extension of G by A, we associate a cohomology class in $H^2(BG;A)$. We prove this association is injective, and bijective in many cases. A homomorphism of such groups lifts to a central extension iff the pullback of the associated cohomology class vanishes.

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arXiv2022

Stiefel-Whitney Classes of Representations of $\text{SL}(2,q)$

We describe the Stiefel-Whitney classes (SWCs) of orthogonal representations $\pi$ of the finite special linear groups $G=\text{SL}(2,\mathbb F_q)$, in terms of character values of $\pi$. From this calculation, we can answer interesting questions about SWCs of $\pi$. For instance, we determine the subalgebra of $H^*(G,\mathbb Z/2\mathbb Z)$ generated by the SWCs of orthogonal $\pi$, and we also determine which $\pi$ have nontrivial mod $2$ Euler class.

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arXiv2021

A Chinese Remainder Theorem for Partitions

Let $s,t$ be natural numbers, and fix an $s$-core partition $\sigma$ and a $t$-core partition $\tau$. Put $d=\gcd(s,t)$ and $m= lcm(s,t)$, and write $N_{\sigma, \tau}(k)$ for the number of $m$-core partitions of length no greater than $k$ whose $s$-core is $\sigma$ and $t$-core is $\tau$. We prove that for $k$ large, $N_{\sigma, \tau}(k)$ is a quasipolynomial of period $m$ and degree $\frac{1}{d}(s-d)(t-d)$.

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arXiv2019

Spinorial Representations of Symmetric Groups

A real representation $\pi$ of a finite group may be regarded as a homomorphism to an orthogonal group $\Or(V)$. For symmetric groups $S_n$, alternating groups $A_n$, and products $S_n \times S_{n'}$ of symmetric groups, we give criteria for whether $\pi$ lifts to the double cover $\Pin(V)$ of $\Or(V)$, in terms of character values. From these criteria, we compute the second Stiefel-Whitney classes of these representations.

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arXiv2019

On the Divisibility of Character Values of the Symmetric Group

Fix a partition $\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let $\chi^\lambda_\mu$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type $(\mu_1,\dotsc,\mu_m,1^{n-k})$. We show that the proportion of partitions $\lambda$ of $n$ such that $\chi^\lambda_\mu$ is divisible by $d$ approaches $1$ as $n$ approaches infinity.

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arXiv2019

Spinoriality of orthogonal representations of reductive groups

Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of the irreducible constituents of the representation.

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arXiv2018

Macdonald trees and determinants of representations for finite Coxeter groups

Every irreducible odd dimensional representation of the $n$'th symmetric or hyperoctahedral group, when restricted to the $(n-1)$'th, has a unique irreducible odd-dimensional constituent. Furthermore, the subgraph induced by odd-dimensional representations in the Bratteli diagram of symmetric and hyperoctahedral groups is a binary tree with a simple recursive description. We survey the description of this tree, known as the Macdonald tree, for symmetric groups, from our earlier work. We describe analogous results for hyperoctahedral groups. A partition $\lambda$ of $n$ is said to be chiral if the corresponding irreducible representation $V_\lambda$ of $S_n$ has non-trivial determinant. We review our previous results on the structure and enumeration of chiral partitions, and subsequent extension to all Coxeter groups by Ghosh and Spallone. Finally we show that the numbers of odd and chiral partitions track each other closely.

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arXiv2017

Determinants of Representations of Coxeter Groups

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial multiplicative character $\omega$ of $W$, we give a closed formula for the number of irreducible representations of $W$ with determinant $\omega$. For Coxeter groups of type $B_n$ and $D_n$, this is accomplished by characterizing the bipartitions associated to such representations.

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arXiv2016

Representations of symmetric groups with non-trivial determinant

We give a closed formula for the number of partitions $\lambda$ of $n$ such that the corresponding irreducible representation $V_\lambda$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.

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arXiv2016

Towards a Goldberg-Shahidi pairing for classical groups

Let G be either an orthogonal, a symplectic or a unitary group over a local field F and let P = MN be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W less than or equal to dim X, excluding only the symplectic case with dim W odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of G. Namely, one obtains a bilinear pairing on matrix coefficients in the spirit of Goldberg-Shahidi, which detects the presence of poles of these operators at 0.

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