Projective representations of symmetric groups via Sergeev duality
Projective representations of symmetric groups via Sergeev duality
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DOI：
10.1007/s002090100282
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发表时间：
2002
影响因子：
0.8
通讯作者：
Jonathan Brundan;A. Kleshchev
中科院分区：
文献类型：
--
作者：
Jonathan Brundan;A. Kleshchev
In this article, we determine the irreducible projective representations of the symmetric group Sd and the alternating group Ad over an algebraically closed field of characteristic p= 2. These matters are well understood in the case p= 0, thanks to the fundamental work of Schur [24] in 1911, as well as the much more recent work of Nazarov [19, 20], Sergeev [25, 26] and others. So the focus here is primarily on the case of positive characteristic, where surprisingly little is known as yet. In particular, we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set 勿乡 p (d) of restricted p-strict partitions of d. Such partitions arose recently in work of Kashiwara, Miwa, Peterson and Yung [11] and Leclerc and Thibon [14] on the q-deformed Fock space of the affine Kac-Moody algebra of type A (2) p− 1. Leclerc and Thibon proposed that 勿乡 p (d) should label the irreducible projective representations in some natural way, and we show here how this can be done. Note that for p= 3, 5, the labelling problem was solved in [1, 3], while if p= 2 all projective representations of Sd and Ad are linear so do not need to be considered further here. To be more precise, recall that λ is a partition of d if λ=(λ1, λ2,...) is a non-increasing sequence of non-negative integers summing to d. Call λ p-strict if in addition λi= λi+ 1⇒ p| λi for each i= 1, 2,....Let 乡 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. Call λ∈ 乡 p (d) a restricted p-strict partition if either p= 0, or p> 0 and