On the (super)cocenter of Cyclotomic Sergeev algebras
Shuo Li, Lei Shi
TL;DR
We study the cyclotomic Sergeev algebra $\mathfrak{H}_n^{g}$, establishing that it is symmetric when the level $d$ is odd and supersymmetric when $d$ is even. A Frobenius-type form $t_{n,d}$ is constructed via a chain of projections, with explicit parity-dependent properties. We prove an integral basis for the degree-zero cocenter $\mathrm{Tr}(\mathfrak{H}_n^{g})_{\overline{0}}$ and identify its rank in terms of colored semi-bipartitions, recovering Ruff's center rank in the odd-level case. Additionally, we construct a generating set for the supersymmetric cocenter $\mathrm{SupTr}(\mathfrak{H}_n^{g})_{\overline{0}}$ and derive an upper bound on the center dimension when $d$ is even, with a complete result at $d=1$ and conjectural rank in general. The results connect cocenter/center dualities in superalgebras to combinatorial parametrizations from complex reflection groups, providing tools for semisimple character theory and potential categorifications.
Abstract
We show that cyclotomic Sergeev algebra $\mathfrak{h}_n^g$ is symmetric when the level is odd and supersymmetric when the level is even. We give an integral basis for ${\rm Tr}(\mathfrak{h}_n^g)_{\overline{0}}$, and recover Ruff's result on the rank of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is odd. We obtain a generating set of ${\rm SupTr}(\mathfrak{h}_n^g)_{\overline{0}}$, which gives an upper bound of the dimension of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is even.