Hecke algebras and immanants
Let \(R\) be an algebra and let \(e\in R\) be an idempotent. The algebra \(eRe\) is called the Hecke algebra associated with \(e\). The author discusses the case when \(R=\mathbb{C} G\); the group algebra of a finite group \(G\) over \(\mathbb{C}\) and \(e={\psi(1)\over|H|}\sum_{h\in H}\psi(h^{-1})h\) where \(H\leq G\) and \(\psi\) is a character of \(H\). Every irreducible character \(\chi\) of \(\mathbb{C} G\) can be restricted to \(H=e\mathbb{C} Ge\) to give (if non-zero) an irreducible character of \(H\). Conversely, every irreducible character of \(H\) is the restriction of some irreducible character of \(\mathbb{C} G\). The author describes how character values of \(H=e\mathbb{C} Ge\) can be used to give linear combinations of immanants which are positive on all Hermitian positive semidefinite matrices \(A\). As an application, a short proof is given of a theorem of \textit{G. D. James} and \textit{M. W. Liebeck} [Proc. Lond. Math. Soc., III. Ser. 55, 243-265 (1987; Zbl 0657.15006)] that the permanent of \(A\) is greater than the normalized immanants which correspond to two-part partitions.
Hermitian positive semidefinite matrices
scientific article; zbMATH DE number 553980
scientific article