A new approach to the representation theory of the symmetric groups. IV: \(\mathbb{Z}_2\)-graded groups and algebras; projective representations of the group \(S_n\). (English) Zbl 1196.20017

Vershik, A. M.; Sergeev, A. N.

Mosc. Math. J. 8, No. 4, 813-842 (2008); corrigendum ibid. 18, No. 1, 187 (2018).

Summary: We start with definitions of the general notions of the theory of \(\mathbb{Z}_2\)-graded algebras. Then we consider theory of inductive families of \(\mathbb{Z}_2\)-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers by A. M. Vershik and A. Okounkov [part I, Sel. Math., New Ser. 2, No. 4, 581-605 (1996; Zbl 0959.20014), part II, Zap. Nauchn. Semin. POMI 307, 57-98 (2004); translation in J. Math. Sci., New York 131, No. 2, 5471-5490 (2005; Zbl 1083.20502)] to representation theory of symmetric groups. The main example is the theory of the projective representations of symmetric groups.
For part III see A. M. Vershik, Mosc. Math. J. 6, No. 3, 567-585 (2006; Zbl 1152.20013).

MSC:

20C30	Representations of finite symmetric groups
05E10	Combinatorial aspects of representation theory
20C25	Projective representations and multipliers
17A70	Superalgebras
16W50	Graded rings and modules (associative rings and algebras)

Keywords:

chains of graded algebras; Gelfand-Tsetlin superalgebras; Young formulas; projective representations

Citations:

Zbl 0959.20014; Zbl 1083.20502; Zbl 1152.20013