A New Proof of the Mullineux Conjecture

By Jonathan Brundan, Jonathan R. Kujawa. 2002

122 citations recorded by the source corpus.

Let Sd denote the symmetric group on d letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible Sd-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchev's approach. Applying the representation theory of the supergroup GL(m | n) and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineux's conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of GL(m | n) of degree d for arbitrary m, n, and d.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c5da0c538e9afd76aecf0f91e11c0717a32af309

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup (2009, background): “it lifts in a unique way to a Gmodule; see e.g. [ BK , Corollary 3.5].”
Grothendieck rings of basic classical Lie superalgebras (2007, background): “[7]) we see that J(g)0,0 is a subring of the Grothendieck ring of this supergroup.”
Chevalley Supergroups (2008, background): “[3], [4], [25] in the super context) we can define the superalgebra of distributions Dist (G) on any supergroup G .”
Representation theory of symmetric groups and related Hecke algebras (2009, unknown): “We refer the reader to [27, 17, 197, 43, 139, 64, 87, 163, 19, 16, 90, 65, 183] as well as [40, Remarks 3.”