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Jason Fulman

Publications and source records attributed to Jason Fulman.

At least 37 records · Page 2Linked to original sources
arXiv2017

Asymptotics of the number of involutions in finite classical groups

Answering a question of Geoff Robinson, we compute the large n limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of involutions in GL(n,q). We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new "sum=product" identities. Our self-contained treatment of the enumeration of involutions in even characteristic symplectic and orthogonal groups may also be of interest.

math.GR↗
arXiv2016

Enumeration of Commuting Pairs in Lie Algebras over Finite Fields

Feit and Fine derived a generating function for the number of ordered pairs of commuting n by n matrices over the finite field F_q. This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for the Lie algebras of the finite general linear and unitary groups, and of odd characteristic symplectic groups.

math.CO↗
arXiv2015

Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representations.

math.GR↗
arXiv2015

On the distribution of the number of fixed vectors for the finite classical groups

Motivated by analogous results for the symmetric group and compact Lie groups, we study the distribution of the number of fixed vectors of a random element of a finite classical group. We determine the limiting moments of these distributions, and find exactly how large the rank of the group has to be in order for the moment to stabilize to its limiting value. The proofs require a subtle use of some q-series identities. We also point out connections with orthogonal polynomials.

math.CO↗
arXiv2015

Stein's method and the rank distribution of random matrices over finite fields

With ${\mathcal{Q}}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_q$ of size $q\ge2$, and ${\mathcal{Q}}_q$ the distributional limit of ${\mathcal{Q}}_{q,n}$ as $n\rightarrow\infty$, we apply Stein's method to prove the total variation bound $\frac{1}{8q^{n+m+1}}\leq\|{\mathcal{Q}}_{q,n}-{\mathcal{Q}}_q\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}$. In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.

math.PR↗
arXiv2015

Derangements in Subspace Actions of Finite Classical Groups

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

math.GR↗
arXiv2014

Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians of random graphs

Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random partitions. We connect these random partitions to the Hall-Littlewood polynomials of symmetric function theory, and use this connection to give combinatorial proofs of properties of these random partitions. In addition, we use Markov chains to give an algorithm for generating these partitions.

math.CO↗
arXiv2013

Combinatorics of balanced carries

We study the combinatorics of addition using balanced digits, deriving an analog of Holte's "amazing matrix" for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be 1,1/b,...,1/b^n, and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes characters, and that the right eigenvectors can be identified with hyperoctahedral Eulerian idempotents. We also examine the carries that occur when a column of balanced digits is added, showing this process to be determinantal. The transfer matrix method and a serendipitous diagonalization are used to study this determinantal process.

math.CO↗
arXiv2013

Analysis of casino shelf shuffling machines

Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed.

math.CO↗
arXiv2013

Cohen-Lenstra heuristics and random matrix theory over finite fields

Let g be a random element of a finite classical group G, and let λ_{z-1}(g) denote the partition corresponding to the polynomial z-1 in the rational canonical form of g. As the rank of G tends to infinity, λ_{z-1}(g) tends to a partition distributed according to a Cohen-Lenstra type measure on partitions. We give sharp upper and lower bounds on the total variation distance between the random partition λ_{z-1}(g) and the Cohen-Lenstra type measure.

math.NT↗
arXiv2013

Generating functions for real character degree sums of finite general linear and unitary groups

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or -1.

math.GR↗
arXiv2012

Exponential approximation and Stein's method of exchangeable pairs

We derive a new result for exponential approximation using Stein's method of exchangeable pairs. As an application, an exponential limit theorem with error term is derived for |Tr(U)|^2, where Tr(U) denotes the trace of a matrix chosen from the Haar measure of the unitary group U(n,C).

math.PR↗
arXiv2011

Stein's method, heat kernel, and linear functions on the orthogonal groups

Combining Stein's method with heat kernel techniques, we study the function Tr(AO), where A is a fixed n by n real matrix over such that Tr(AA^t)=n, and O is from the Haar measure of the orthogonal group O(n,R). It is shown that the total variation distance of the random variable Tr(AO) to a standard normal random variable is bounded by 2 * squareroot(2) /(n-1), slightly improving the constant in a bound of Meckes, which was obtained by completely different methods.

math.PR↗
arXiv2011

Zero biasing and growth processes

The tools of zero biasing are adapted to yield a general result suitable for analyzing the behavior of certain growth processes. The main theorem is applied to prove central limit theorems, with explicit error terms in the L^1 metric, for certain statistics of the Jack measure on partitions and for the number of balls drawn in a Polya-Eggenberger urn process.

math.CO↗
arXiv2011

Foulkes Characters, Eulerian Idempotents, and an Amazing Matrix

John Holte [16] introduced a family of "amazing matrices" which give the transition probabilities of "carries" when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra [4,6,7] and in the analysis of riffle shuffling [6,7]. We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday [20] in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.

math.CO↗