Invariants of GLn(𝔽q) in polynomials modulo Frobenius powers

Lewis, Joel Brewster; Reiner, Victor; Stanton, Dennis

doi:10.1017/s0308210516000366

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…. , θ n span a GL n (F q )-stable subspace over F q with the map x i → x q m i defining a GL n (F q )-equivariant isomorphism (see [9]). The quotient S/m [q m ] is (q m ) n -dimensional, and Lewis, Reiner, and Stanton give a conjecture for the Hilbert series of its GL n (F q )-fixed subspace:…”

Section: Motivationmentioning

confidence: 99%

“…Orbits and the dimension of the invariant space. The conjecture of Lewis, Reiner, and Stanton [9] giving the Hilbert series for the GL n (F q )-invariants in S/m [q m ] specializes to a conjecture for the dimension of the invariants as an F q -vector space. They show this specialization gives the number of orbits for GL n (F q ) acting on the vector space V ′ = (F q m ) n , see [9, Section 7.1 and Theorem 6.16].…”

Section: Full Pointwise Stabilizer In the General Linear Groupmentioning

confidence: 99%

“…In 2017, Lewis, Reiner and Stanton [9] conjectured a combinatorial formula for the Hilbert series of a space of invariants under the action of the general linear group GL n (F q ) over a finite field F q in terms of (q, t)-binomial coefficients. This formula provides an analogue for the q-Catalan and q-Fuss Catalan numbers that connect Hilbert series for certain invariant spaces with the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups.…”

Section: Introductionmentioning

confidence: 99%

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Invariants of polynomials mod Frobenius powers

Drescher

Shepler

2020

Journal of Algebra

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Lewis, Reiner, and Stanton conjectured a Hilbert series for a space of invariants under an action of finite general linear groups using (q, t)-binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan numbers to the representation theory of rational Cherednik algebras. They consider a finite general linear group as a reflection group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. We prove a variant of their conjecture in the local case, when the group acting fixes a reflecting hyperplane, over fields of prime order. m [q m ] := (x q m 1 , . . . , x q m n ) , which we call the Frobenius irrelevant ideal. Their conjecture gives the Hilbert series for the GL n (F q )-invariants in F q [x 1 , . . . , x n ]/(x q m 1 , . . . , x q m n ) using (q, t)-binomial coefficients. We consider subgroups of reflections about a single hyperplane H in V . These groups are not cyclic in general, in contrast to groups over fields of characteristic 0. We take the case when q is a prime p; some of our ideas generalize to arbitrary q. We explicitly describe the space of G-invariants in S/m [p m ] for any subgroup G ⊂ GL n (F p ) fixing a hyperplane H in V pointwise. We give the Hilbert series in terms of the dimension of the transvection root space. As a special case, we describe the invariants under the pointwise stabilizer GL n (F p ) H in GL n (F p ) of any hyperplane H in V .

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Section: Motivationmentioning

confidence: 99%

Section: Full Pointwise Stabilizer In the General Linear Groupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Invariants of polynomials mod Frobenius powers

Drescher

Shepler

2020

Journal of Algebra

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“…(cf. [5] Proposition B.14) Let R be an N-graded ring. Let I ⊂ R + := ⊕ d>0 R d be a homogeneous ideal of positive degree elements.…”

Section: Furthermore We Havementioning

confidence: 99%

Torsions in Cohomology of $\text{SL}_2(\mathbb{Z})$ and Congruence of Modular Forms

Deng

2019

Preprint

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“…When |G| is not a unit, very little is known about S G as an S G -module. In [8], Lewis, Reiner, and Stanton prove that S G still has rank one over S G , and they give conjectures for the Hilbert series of F q [x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Symmetric group fixed quotients of polynomial rings

Pevzner¹

2023

Preprint

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Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p.

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Invariants of GL_n(𝔽_q) in polynomials modulo Frobenius powers

Abstract: Abstract. Conjectures are given for Hilbert series related to polynomial invariants of finite general linear groups, one for invariants mod Frobenius powers of the irrelevant ideal, one for cofixed spaces of polynomials.

Cited by 6 publications

References 21 publications

Invariants of polynomials mod Frobenius powers

Invariants of polynomials mod Frobenius powers

Torsions in Cohomology of $\text{SL}_2(\mathbb{Z})$ and Congruence of Modular Forms

Symmetric group fixed quotients of polynomial rings

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