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On the denominators of Young’s seminormal basis. (English) Zbl 07559760

Summary: We study the seminormal basis \(\{ f_{\mathfrak{t}} \}\) for the Specht modules of the Iwahori-Hecke algebra \(\mathcal{H}_n(q)\) of type \(A_{n - 1}\). We focus on the base change coefficients between the seminormal basis \(\{f_{\mathfrak{t}}\}\) and Murphys’ standard basis \(\{x_{\mathfrak{t}}\}\) with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving hook lengths. Even for general standard tableaux we obtain new formulas. On the way we prove a new result about submodules of the restricted Specht module at root of unity.

MSC:

20C08 Hecke algebras and their representations
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory

References:

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