I am looking for the character table of the Sergeev group S_d for small d (say, 'd' up to 10 or up to whatever is possible).
The Sergeev group $S_d$ is defined as follows:
- Let $\mathfrak{S}_d$ be the symmetric group on d elements
- Let $\text{Cliff}_d$ be finite group generated by elements $x_1, ..., x_d,\epsilon$ such that $x_i^2 = \epsilon^2 = 1$ and $x_i x_j = \epsilon x_j x_i$ for $i \neq j$.
- Define $S_d := \mathfrak{S}_d \ltimes \text{Cliff}_d$, where the action permutes the generators $x_i$.
I appreciate explicit tables as answers, already for d=3 or higher. Alternatively as a plan B an indication of a software and how to compute them (I guess that I can probably spend quite some time installing+learning coding in Magma or Gap (?) and produce these tables myself, but the hope is that some good-soul group theorist for which this takes only a few minutes instead could help out).
Thanks a lot in advance!
Best, Dani
Recently, I happened to stumble upon the same question. It seems that a description of the character table of the Sergeev group is not available in the literature, although one can find a lot of closely related results. In fact, with a bit of work one can deduce this table from the work of Nazarov. Following among other sources the excellent review on the (super)representations of the Sergeev group you wrote after raising this question, we described the character table of the Sergeev group in a recent paper on area Siegel–Veech constants (see below). In particular, in the appendix, one can find the character table for $d\leq 5$.
Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127, No. 2, 190-257 (1997). ZBL0930.20011.
Alessandro Giacchetto, Reinier Kramer and Danilo Lewański, A new spin on Hurwitz theory and ELSV via theta characteristics, preprint, 56 pages (2021).
Jan-Willem van Ittersum and Adrien Sauvaget, Cylinder counts and spin refinement of area Siegel–Veech constants, preprint, 42 pages (2022).