By Alexander Kleshchev
The illustration conception of symmetric teams is among the most lovely, renowned, and demanding elements of algebra with many deep relatives to different parts of arithmetic, corresponding to combinatorics, Lie conception, and algebraic geometry. Kleshchev describes a brand new method of the topic, in line with the hot paintings of Lascoux, Leclerc, Thibon, Ariki, Grojnowski, Brundan, and the writer. a lot of this paintings has basically seemed within the study literature sooner than. even though, to make it available to graduate scholars, the idea is built from scratch, the one prerequisite being a typical path in summary algebra. Branching ideas are inbuilt from the outset leading to a proof and generalization of the hyperlink among modular branching ideas and crystal graphs for affine Kac-Moody algebras. The equipment are merely algebraic, exploiting affine and cyclotomic Hecke algebras. For the 1st time in booklet shape, the projective (or spin) illustration concept is taken care of alongside an identical traces as linear illustration thought. the writer is especially focused on modular illustration thought, even if every thing works in arbitrary attribute, and in case of attribute zero the technique is a bit of just like the idea of Okounkov and Vershik, defined right here in bankruptcy 2. For the sake of transparency, Kleshschev concentrates on symmetric and spin-symmetric teams, even though the equipment he develops are fairly normal and observe to a few comparable gadgets. In sum, this special ebook might be welcomed through graduate scholars and researchers as a latest account of the topic.
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Extra resources for Linear and Projective Representations of Symmetric Groups
Now fix some total order ≺ refining the Bruhat order < on D x ∈ D , set x ≺x x = y∈D y x y∈D y≺x = = x/ . 1 that x and ≺x are invariant under right = , we have defined a filtration of multiplication by n . Hence, since n as an -bimodule. We want to describe the quotients x explicitly. n To this end, note using the property (2) above that for each y ∈ D , there exists an algebra isomorphism y−1 ∩y → y−1 ∩ Degenerate affine Hecke algebra 30 with y−1 w = y−1 wy and y−1 xi = xy−1 i for w ∈ S ∩y , 1 ≤ i ≤ n.
The main difference however is that n is not semisimple, so we have to consider generalized eigenspaces, rather than usual eigenspaces. We define the formal character of an n -module M as the generating function for the dimensions of simultaneous generalized eigenspaces of the elements xn on M. In Chapter 5 we will prove that the formal characters of x1 irreducible n -modules are linearly independent (as any reasonable formal characters should be). The “Shuffle Lemma”, which is a special case of the Mackey Theorem, gives a transparent description of what induction “does” to the formal characters.
1, this implies that N = 1 ⊗ L. This shows that M contains 1 ⊗ L, but 1 ⊗ L generates the whole of L an over n n is the only irreducible in its block use n . So M = L a . 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.
Linear and Projective Representations of Symmetric Groups by Alexander Kleshchev