Noncommutative invariants of finite and classical groups

2025/02/23 by Ganapathy, Karthik
#05E05 #13A50 #16P90 #16W22 (Primary) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2502.16675

Abstract

We investigate the structure of the invariant subring of the tensor algebra T(W) of a G-representation W, viewed as a twisted commutative algebra (tca). For a faithful representation W of a finite group G over a field k, we show that if char(k) | #G, then T(W)^G is not finitely generated as a tca. In contrast, for a representation W of a classical group G_ℤ, we prove that the invariant subring T(W_k)^G_k is finitely generated as a tca when k is algebraically closed of sufficiently large characteristic, provided that W admits a good filtration over ℤ. Finally, we introduce a categorical variant of the Gelfand--Kirillov dimension and compute its value to be \binomn+12 for T(ℂⁿ) as a tca. Our key insight is to use the Schur functor to reduce questions about noncommutative invariants to those concerning vector invariants.

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