A q-analog of transposition symmetry for invariant G-functions
We give an algebraic construction of Milne's \(^{\nu}_{\mu}[G]_ m^{(n)}\) (\(\gamma\) ;\(\delta\) ;z) functions in terms of the matrix coefficients of certain linear operators \(B_{\mu z}\). We show that the \(^{\nu}_{\mu}[G]_ m^{(n)}\) (\(\gamma\) ;\(\delta\) ;z) functions all satisfy ''transposition'' symmetry if and only if \(z=(t,...,t)\in {\mathbb{C}}^ n\), unless \(^{\nu}_{\mu}[G]_ 1^{(n)}\) (\(\gamma\) ;\(\xi\) ;z) is identically zero. This transposition symmetry is a ''q-analog'' of that for the ordinary \(^{\nu}_{\mu}G_ m^{(n)}\) (\(\gamma\) ;\(\delta)\) polynomials of Biedenharn, Gustafson, and Milne. We also define a ''q-analog'' of the elementary reduced Wigner coefficients for U(n) and give both a combinatorial and representation theoretic interpretation of this definition.
scientific article; zbMATH DE number 3941957
scientific article