对称群的Generalized characters
Generalized characters of the symmetric group
摘要：
对称群S（n）的正规化不可约特征标可以理解为Gelfand对$(S(n)\times S(n),\Diag S(n))$的带状球函数。它们在群S（n）上的函数空间中形成一个关于S（n）的共轭不变的正交基。在本文中，我们考虑一个不同的Gelfand对连接到对称群，这是一个“不平衡”的Gelfand对$(S(n)\times S(n-1),\Diag S(n-1))$。这个Gelfand对的带状球面函数在S（n）上的一个更大的函数空间中形成一个正交基，即在$S(n-1)$共轭不变的函数空间中。我们把这些带状球函数称为S（n）的归一化\textit{generalized}特征标。本文的主要发现是这些广义特征标可以在与对称群的不可约特征标相同的水平上计算。本文给出了S（n）的广义特征标的Murnaghan-Nakayama型定则、Frobenius型公式和行列式公式的类似公式。
英文摘要：
Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair $(S(n)\times S(n),\Diag S(n))$. They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an ``unbalanced'' Gelfand pair $(S(n)\times S(n-1),\Diag S(n-1))$. Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by $S(n-1)$. We refer to these zonal spherical functions as normalized \textit{generalized} characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).
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