Subspaces where an immanant is convertible into its conjugate^∗

^∗This work was done within the activities of the Centro de Estruturas Lineares e Combinatórias and was supported by FundaÇão para a Ciência e Tecnologia and Programa Ciência, Tecnologia e InovaÇão do Quadro Comunitário de Apoio

Abstract

Let n≥5 and let be an irreducible nonlinear character of S_n such that whenever σ is a transposition or a cycle of length three; furthermore let T_n be the (0, 1)-matrix of order n that has ones exactly on and below the upper neighbours of the main diagonal and denote by E_ij the matrix of order n with 1 in position (i, j) and 0 elsewhere.

Given i,jε{1,…,n}, with i+1<j, we prove that if j−i≠3, then in the subspace M_n (T_n +E_ij there exist matrices for which the immanant is not convertible into the immanant by sign-affixing. Abusing language, we say that the space is -inconvertible, and show that spaces M_n (T_n +E₂₅ ) and M_n (T_n +E_n−3,n ). We also state some sufficient fonditions for the subspace M_n (T_n ) to be external convertible.

With some exceptions our theorems say that the coordinate subspaces found for the conversion of the permanent into the determinant by Gibson around 1970 are also best possible for other immanants.

Keywords:

• Immanants
• AMS Subject Classification: 15 A 15

^∗This work was done within the activities of the Centro de Estruturas Lineares e Combinatórias and was supported by FundaÇão para a Ciência e Tecnologia and Programa Ciência, Tecnologia e InovaÇão do Quadro Comunitário de Apoio

^∗This work was done within the activities of the Centro de Estruturas Lineares e Combinatórias and was supported by FundaÇão para a Ciência e Tecnologia and Programa Ciência, Tecnologia e InovaÇão do Quadro Comunitário de Apoio

Notes

^∗This work was done within the activities of the Centro de Estruturas Lineares e Combinatórias and was supported by FundaÇão para a Ciência e Tecnologia and Programa Ciência, Tecnologia e InovaÇão do Quadro Comunitário de Apoio