Characterization of characters and pseudo-characters
Let \(R\) be an algebra over a commutative ring \(A\). An \(A\)-linear functional \(f:R\to A\) is called central if \(f(xy)=f(yx)\) for all \(x,y\in R\). For a central functional \(f\), a map \(S_n(f):R\to A\) is defined by \(S_n(f)(x)=\sum_{\sigma\in{\mathfrak S}_n}\varepsilon(\sigma)(\sigma\cdot f)(x)\), where \({\mathfrak S}_n\) is the symmetric group on \(\{1,\dots,n\}\), \(\varepsilon(\sigma)\) is the signature of the permutation \(\sigma\), and, if \(\sigma\) decomposes into a product of disjoint cycles \(\sigma=\sigma_1\dots\sigma_k\) with \(\sigma_i=(j_1,\dots,j_m)\), then \((\sigma\cdot f)(x)=f(\sigma_1(x))\dots f(\sigma_k(x))\) where \(f(\sigma_i(x))=f(x_{j_1}\dots x_{j_m})\) if \(x=(x_1,\dots,x_n)\). For \(d\) a non-negative integer, a central functional \(f\) is called a pseudo-character of dimension \(d\) if \(S_{d+1}(f)=0\) and \(S_k(f)\neq 0\) for \(k\leq d\). The author shows that the reduced trace of an Azumaya algebra is a pseudo-character, and that therefore characters are pseudo-characters. If the base ring \(A\) is a field, the author uses the reduced trace to set up to a one-to-one correspondence between absolutely irreducible representations in a central simple algebra of degree \(n\) and absolutely irreducible pseudo-characters of dimension \(n\). This correspondence is generalized to establish a bijection between absolutely irreducible representations in Azumaya algebras of constant rank \(n^2\) and pseudo-characters of dimension \(n\) whose residue pseudo-characters are absolutely irreducible of dimension \(n\).
absolutely irreducible pseudo-characters
scientific article; zbMATH DE number 869794