A proof of the Lewis-Reiner-Stanton conjecture for the Borel subgroup

Abstract

For each parabolic subgroup P \mathrm {P} of the general linear group GL n ⁡ ( F q ) \operatorname {GL}_n(\mathbb {F}_q) , a conjecture due to Lewis, Reiner and Stanton [Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), pp. 831–873] predicts a formula for the Hilbert series of the space of invariants Q m ( n ) P \mathcal {Q}_m(n)^\mathrm {P} where Q m ( n ) \mathcal {Q}_m(n) is the quotient ring F q [ x 1 , … , x n ] / ( x 1 q m , … , x n q m ) \mathbb {F}_q[x_1,\ldots ,x_n]/(x_1^{q^m},\ldots ,x_n^{q^m}) . In this paper, we prove the conjecture for the Borel subgroup B \mathrm {B} by constructing a linear basis for Q m ( n ) B \mathcal {Q}_m(n)^\mathrm {B} . The construction is based on an operator δ \delta which produces new invariants from old invariants of lower ranks. We also upgrade the conjecture of Lewis, Reiner and Stanton by proposing an explicit basis for the space of invariants for each parabolic subgroup.