On the representation theory of the symmetric groups

The theory of the covering groups S̃n of the symmetric groups Sn has its origins in early papers by Schur [S1,S2,S3] who investigated the projective representations of the groups Sn. While the representation theory of the symmetric groups made continuous progress, with the appropriate combinatorial concepts of partitions and tableaux developed early on by Young, the representation theory of the covering groups saw a long period of stagnation. Only 50 years after Schur’s fundamental paper on the group S̃n, new light into the subject was brought by Morris [Mo1, Mo2]. He introduced the combinatorial notions which turned out to be the right analogues of the notions in the Sn case: shifted tableaux instead of Young tableaux and bars instead of hooks. Using these notions he obtained new results on the Q-functions introduced by Schur, and he found a recursive formula for the irreducible spin characters (i.e. the characters belonging to the proper projective representations) in analogy to the Murnaghan-Nakayama formula, thus opening up a deeper investigation of the characters of S̃n. In the meantime, p-modular representation theory of finite groups had been developed, mainly by Richard Brauer, who had defined the fundamental concept of p-blocks of finite groups. While the p-blocks of the groups Sn had been determined already in the 1940’s by a combinatorial algorithm – still known as the Nakayama conjecture – it was only in 1965 that Morris conjectured how to determine the p-blocks of S̃n for odd primes p, again by a combinatorial procedure. It took more than 20 years before this conjecture was settled by Humphreys [H] and then, using different methods, by Cabanes [C]. In the meantime, there was very little progress on the modular representation theory of the groups S̃n. But in the past decade the area has seen many significant contributions, not only from representation theory, but now also from combinatorialists interested in tableaux (see [St1, St2]). At about the

Representations of the covering groups of the symmetric groups and their combinatorics.

/pdf/review-gordon-james-and-adalbert-kerber-the-representation-2symnndcwj.pdf

Review: Gordon James and Adalbert Kerber, The representation theory of the symmetric group

Algebra and Combinatorics

We propose a generalization of Schur regular partitions for each odd integer $p\geq 3$. Applying Kashiwara crystal theory, we prove that the number of partitions of $n$ with this condition is equinumerous to the number of strict $p$-class regular partitions of $n$. At $p=3$, it is Schur's 1926 partition theorem found as a mod 6 analog of the Rogers-Ramanujan partition theorem (RRPT). The statement for $p=5$ was conjectured by Andrews in 1970s in a course of his 3 parameter generalization of RRPT and proved in 1994 by Andrews-Bessenrodt-Olsson with an aid of computer.

Schur partition theorems via perfect crystal

The modular characters of the symmetric groups

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The modular characters of the symmetric groups