Abstract
Let σ 1 = 1, σ 2 = (12), σ 3 = (12)(34),… be the conjugacy classes of involutions in S k . It was shown by Klyachko and by Inglis et al. [82] that it is possible to specify a set of characters \(\psi _{1},\psi _{2},\psi _{3},\ldots\) of degree 1 of the centralizers of \(\sigma _{1},\sigma _{2},\sigma _{3},\ldots\) such that the direct sum of the induced representations of the \(\psi_i\) contains every irreducible representation exactly once. In the next chapter, we will see that translating this fact and related ones to the unitary group gives classical facts about symmetric and exterior algebra decompositions due to Littlewood [120].
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Bump, D. (2013). The Involution Model for S k . In: Lie Groups. Graduate Texts in Mathematics, vol 225. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8024-2_43
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DOI: https://doi.org/10.1007/978-1-4614-8024-2_43
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