Explicit Hilbert–Kunz functions of $2 \times 2$ determinantal rings

Marcus Robinson and Irena Swanson

Vol. 275 (2015), No. 2, 433–442

DOI: 10.2140/pjm.2015.275.433

Abstract

Let $k\left[X\right]=k\left[x_{i,j}:i=1,\dots ,m;j=1,\dots ,n\right]$ be the polynomial ring in $mn$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2\left(X\right)$ the ideal generated by the $2\times 2$ minors of the generic $m\times n$ matrix $\left[x_{i,j}\right]$. We give a closed polynomial formulation for the dimensions of the $k$-vector space $k\left[X\right]∕\left(I_2\left(X\right)+\left(x_{1,1}^q,\dots ,x_{m,n}^q\right)\right)$ as $q$ varies over all positive integers, i.e., we give a closed polynomial form for the generalized Hilbert–Kunz function of the determinantal ring $k\left[X\right]∕I_2\left(X\right)$. We also give a closed formulation of dimensions of other related quotients of $k\left[X\right]∕I_2\left(X\right)$. In the process we establish a formula for the numbers of some compositions (ordered partitions of integers), and we give a proof of a new binomial identity.

Keywords

Hilbert–Kunz function, multiplicity, combinatorial identity

Mathematical Subject Classification 2010

Primary: 13D40

Secondary: 05A15, 05A10

Milestones

Received: 4 March 2014

Revised: 16 December 2014

Accepted: 17 December 2014

Published: 15 May 2015