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Invariants of polynomials mod Frobenius powers. (English) Zbl 1455.20027

Let \(G\) be the general linear group \(\mathrm{GL}_n(\mathbb{F}_q)\) over the finite field \(\mathbb{F}_q\) acting via linear substitutions of variables on the polynomial ring \(S=\mathbb{F}_q[x_1,\dots,x_n]\) and let \(\mathfrak{m}^{[q^k]}\) be the iterated Frobenius powers \((x_1^{q^k},\dots,x_n^{q^k})\) of the maximal ideal \((x_1,\dots,x_n)\) for \(k \in \{ 0,1,\dots, n \}\).
J. B. Lewis et al. [Proc. R. Soc. Edinb., Sect. A, Math. 147, No. 4, 831–873 (2017; Zbl 1393.13014)] conjectured a combinatorial formula for the Hilbert series of \((S/\mathfrak{m}^{[q^k]})^G\), and for the Hilbert series of \(S_G\), the \(G\)-cofixed space of \(S\). This formula provides an analogue for the \(q\)-Catalan and \(q\)-Fuss Catalan numbers which connect Hilbert series for certain invariant spaces with the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups.
In the paper under review, the authors prove a version of the conjecture in the local case. In particular, the authors focus their interest on the situation in which \(G\) fixes a reflecting hyperplane.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
13A50 Actions of groups on commutative rings; invariant theory

Citations:

Zbl 1393.13014
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References:

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