Projective representations of symmetric groups via Sergeev duality.

*(English)*Zbl 1029.20008We 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)].

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)].

Reviewer: Ali Iranmanesh (Tehran)

##### 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 |