Representation of symmetric functions as Gram determinants
Let \({\mathfrak Q}\) be the complex vector space of all rational functions in a complex variable \(z\). Using an alternating bilinear from \(\langle , \rangle\) introduced earlier [see Adv. Math. 116, No. 2, 377-388 (1995; Zbl 0845.16038)], the author defines a symmetric bilinear form \(S\) on \({\mathfrak Q}^{n+1}\) as follows: if \({\mathbf u}= (u_0,\dots, u_n)\), \({\mathbf v}= (v_0,\dots, v_n)\), then \(S({\mathbf u},{\mathbf v})= [\langle u_j, Rv_k\rangle]\), where \(R\) is the involution on \({\mathfrak Q}\) that sends \(f(z)\) to \(z^{-1}f(z^{-1})\). If, in particular, \(u_j= z^j/w(z)\) \((j= 0,\dots, n)\), where \(w(z)= (z- x_0)\cdots(z- x_n)\), then \(S({\mathbf u},{\mathbf v})\) is a symmetric function of the \(x_i\) for each choice of \({\mathbf v}\). Various classical symmetric functions, such as the \(S\)-functions, arise in this way. As an illustration of the method, two functional relations satisfied by \(S({\mathbf u},{\mathbf v})\) are shown to specialize to the Jacobi-Trudi formula for \(S\)-functions and Cauchy identity for \(\Pi(1- x_iy_j)^{-1}\). Clearly many variations can be played on the author's theme and he makes brief mention of a few of them.
scientific article; zbMATH DE number 1241238