Rook placements and Jordan forms of upper-triangular nilpotent matrices

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of Jordan type lambda as a weighted sum over standard Young tableaux. We also study a connection between these matrices and non-attacking rook placements, which leads to a refinement of the formula for F_\lambda(q).Comment: 25 pages, 6 figure

q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements

q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

Abstract. The set of n by n upper-triangular nilpotent matrices with entries in a finite field Fq has Jordan canonical forms indexed by partitions λ ⊢ n. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number Fλ(q) of matrices of fixed Jordan type as a weighted sum over rook placements. Résumé. L’ensemble des matrices triangulaires supérieures nilpotentes d’ordre n sur un corps fini Fq a des formes canoniques de Jordan indexées par les partitions λ ⊢ n. Nous étudions une connexion entre ces matrices et les placements de tours, et nous présentons une formule combinatoire pour le nombre Fλ(q) des matrices comme une somme sur les placements de tours

q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements

Fast Computation of the NN-th Term of a qq-Holonomic Sequence and Applications

33 pages. Long version of the conference paper Computing the $N$-th term of a $q$-holonomic sequence. Proceedings ISSAC'20, pp. 46–53, ACM Press, 2020 (https://hal.inria.fr/hal-02882885)International audienceIn 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen’s algorithm to the computation of the $N$-th term of any holonomic sequence in essentially the same arithmetic complexity. We design $q$-analogues of these algorithms. We first extend Strassen’s algorithm to the computation of the $q$-factorial of $N$, then Chudnovskys' algorithm to the computation of the $N$-th term of any $q$-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in $\sqrt{N}$; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear $q$-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost

q-Whittaker functions, finite fields, and Jordan forms

The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial $P_\lambda(\mathbf{x};q,t)$. We show combinatorially how to expand $W_\lambda(\mathbf{x};q)$ in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size $1/q$. This yields an expression analogous to a well-known formula for the Hall-Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, proving the Cauchy identity for $q$-Whittaker functions. We call our probabilistic bijection the $q$-Burge correspondence, and prove that in the limit as $q\to 0$, we recover a description of the classical Burge correspondence (also known as column RSK) due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary double coset of $\text{GL}_n$ modulo two parabolic subgroups, which we find to be of independent interest. As an application, we use the $q$-Burge correspondence to count isomorphism classes of certain modules over the preprojective algebra of a type $A$ quiver (i.e. a path), refined according to their socle filtrations. This develops a connection between the combinatorics of symmetric functions and the representation theory of preprojective algebras.Comment: 69 pages. v2: Added Remark 5.1