Hilbert functions of certain rings of invariants via representations of the symmetric groups (with an appendix by Dejan Govc)
| Autor: | Ehud Meir |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Ring (mathematics)
Hilbert series and Hilbert polynomial Pure mathematics Algebra and Number Theory 010102 general mathematics Field (mathematics) 01 natural sciences Representation theory Invariant theory symbols.namesake 0103 physical sciences symbols 010307 mathematical physics 0101 mathematics Vector space Rational representation Hilbert–Poincaré series Mathematics |
| Zdroj: | Journal of Algebra. 572:1-35 |
| ISSN: | 0021-8693 |
| DOI: | 10.1016/j.jalgebra.2020.12.004 |
| Popis: | In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K [ U ] Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U = End ( W ⊕ k ) and Γ = GL ( W ) and on the case where U = End ( V ⊗ W ) and Γ = GL ( V ) × GL ( W ) , though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series, using Mathematica (see the appendix by Dejan Govc). |
| Databáze: | OpenAIRE |
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