Canonical bases related to the symmetric groups

Malka Schaps (Bar-Ilan)

Frank Adams 1,

If we consider the set of all symmetric groups, then the irreducible representations over a field F of characteristic e are labeled by the so-called e-regular partitions.  If we let K_n be the abelian group generated by isomorphism classes of irreducible representations, then certain inductions and restrictions give maps between the K_n, and thus endomorphisms of their direct sum K.  These restrictions and inductions can be viewed as generators e_i and f_i of a certain affine Lie algebras, and K is isomorphic to a particular highest weight representation for this Lie algebra, and, in fact, for its quantum enveloping algebra over a parameter q.  A canonical basis is a basis of this highest weight representation which behaves well with respect to a mapping called the bar involution which sends q to its inverse.  The entire situation can be generalized to other highest weight representations by means of an object called Fock space.

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