# zbMATH — the first resource for mathematics

Projective representations of symmetric groups via Sergeev duality. (English) Zbl 1029.20008
We say that $$\lambda$$ is a partition of $$d$$ if $$\lambda=(\lambda_1,\lambda_2,\dots)$$ is a non-increasing sequence of non-negative integers summing to $$d$$. We say that the partition $$\lambda$$ is $$p$$-strict if in addition $$\lambda_i=\lambda_{i+1}\Rightarrow p\mid\lambda_i$$ for each $$i=1,2,\dots$$. Let $$\mathbb{P}_p(d)$$ denote the set of all $$p$$-strict partitions of $$d$$. Thus, the $$0$$-strict partitions are just the partitions with no repeated non-zero parts, while a $$p$$-strict partition for $$p>0$$ can only have repeated parts if they are divisible by $$p$$. We call $$\lambda\in\mathbb{P}_p(d)$$ a restricted $$p$$-strict partition if either $$p=0$$, or $$p>0$$ and $\begin{cases}\lambda_i-\lambda_{i+1}\leq p\text{ if }p\nmid\lambda_i,\\ \lambda_i-\lambda_{i+1}<p\text{ if }p\mid\lambda_i.\end{cases}$ for each $$i=1,2,\dots$$. Let $$\mathbb{R}\mathbb{P}_p(d)\subseteq\mathbb{P}_p(d)$$ denote the restricted $$p$$-strict partitions of $$d$$.
In the paper under review, the authors determine the irreducible projective representations of the symmetric group $$S_d$$ and the alternating group $$A_d$$ over an algebraically closed field of characteristic $$p\neq 2$$. The construction is based closely on the idea of Sergeev and Nazarov in the characteristic $$0$$ theory. In particular the key step is to determine the irreducible “polynomial” representations of the subgroup $$Q(n)$$ in characteristic $$p$$. These turn out to be labelled naturally according to highest weight theory by all $$p$$-strict partitions with at most $$n$$ non-zero parts. Then Sergeev’s superalgebra analogue of Schur-Weyl duality is used to determine the irreducible representations of a certain twisted version of the group algebra of the hyperoctahedral group. Finally, they pass from there to the symmetric group using methods of M. Nazarov [Adv. Math. 127, No. 2, 190-257 (1997; Zbl 0930.20011)] and A. Sergeev [Represent. Theory 3, No. 14, 416-434 (1999; Zbl 0999.17014)]. Note that for $$p=3,5$$, the labelling problem was already solved by G. E. Andrews, C. Bessenrodt and J. B. Olsson [in Trans. Am. Math. Soc. 344, No. 2, 597-615 (1994; Zbl 0806.05065)] and C. Bessenrodt, A. O. Morris and J. B. Olsson [J. Algebra 164, No. 1, 146-172 (1994; Zbl 0835.20018)].

##### MSC:
 20C25 Projective representations and multipliers 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory
Full Text: