By Alexander Kleshchev
The illustration idea of symmetric teams is likely one of the most lovely, renowned, and critical elements of algebra with many deep kinfolk to different components of arithmetic, similar to combinatorics, Lie idea, and algebraic geometry. Kleshchev describes a brand new method of the topic, in response to the hot paintings of Lascoux, Leclerc, Thibon, Ariki, Grojnowski, Brundan, and the writer. a lot of this paintings has merely seemed within the examine literature earlier than. even though, to make it obtainable to graduate scholars, the speculation is constructed from scratch, the single prerequisite being a regular path in summary algebra. Branching ideas are in-built from the outset leading to an evidence and generalization of the hyperlink among modular branching principles and crystal graphs for affine Kac-Moody algebras. The tools are merely algebraic, exploiting affine and cyclotomic Hecke algebras. For the 1st time in publication shape, the projective (or spin) illustration idea is taken care of alongside an analogous traces as linear illustration thought. the writer is especially occupied with modular illustration idea, even supposing every little thing works in arbitrary attribute, and in case of attribute zero the technique is a little just like the speculation of Okounkov and Vershik, defined the following in bankruptcy 2. For the sake of transparency, Kleshschev concentrates on symmetric and spin-symmetric teams, although the equipment he develops are fairly normal and follow to a few similar gadgets. In sum, this detailed e-book could be welcomed via graduate scholars and researchers as a contemporary account of the topic.
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Additional resources for Linear and Projective Representations of Symmetric Groups
3 There is a natural isomorphism d Hom of n -modules, n for every left n⊗ M M -module M. 2. 10]. 4 There is a natural isomorphism indn M for every finite dimensional Proof The functor indn Hence it is isomorphic to Hom indn d M -module M. -mod → n -mod is right adjoint to resn . n ? by uniqueness of adjoint functors. 3 (with swapped and d replaced by d−1 ). 8 Intertwining elements We will need certain elements of i < n, define i n which go back Cherednik. 22) means that for every w ∈ Sn we obtain a well-defined element w ∈ n , namely, w = i1 im where w = si1 sim is any reduced expression for w.
If is such an orbit we set = a for any a ∈ . Now let M be a finite dimensional n -module and ∈ F n /∼. We let M denote the generalized eigenspace of M over Z n that corresponds to the central character , that is M = v∈M z− z k v = 0 for all z ∈ Z n and k 0 First results on 38 n -modules Observe this is an n -submodule of M. Now, for any a ∈ F n with a ∈ , · · · L an via the central character . 1. decomposes as M= M ∈F n /∼ as an n -module. 2 shows that every such central character does arise in some finite dimensional n -module.
To see that L a Frobenius reciprocity and the fact just proved that L an is irreducible. (ii) The fact that all composition factors of resn L an are isomorphic · · · L a r follows by formal characters and (i). To see that to L a 1 n n ⊗ L of res L an soc res L a is irreducible, note that the submodule · · · L a l . This module is irreducible, and so it is isomorphic to L a 1 -submodule is contained in the socle. Conversely, let M be an irreducible n of L a . 1 as in the proof of (i), we see that M must ⊗ L.