Bilinear forms on modular symmetry classes of tensors (Q1580449)

Let \(F\) be an algebraic closed field of characteristic \(p\) (viewed as the residue field of a valuation ring of the field of algebraic numbers over \({\mathbb Q}\) with its maximal ideal \(\mathfrak{p}\) lying over \(p\)), \(G\) a subgroup of the symmetric group \(S_m\) and \(Bl(G)\) the set of all \(p\)-blocks of \(G\) over \(F\). For any \(B\in Bl(G)\) define the element \[ e^*_B=\sum_{g\in G}\left(\frac{1}{|G|}\sum_{\chi\in \text{Irr}(B)}\chi(1) \chi(g)\right)^* g^{-1}, \] where \(\text{Irr}(B)=B\cap \text{Irr}(G)\) and \((\cdot)^*\) denotes the reduction modulo \(\mathfrak{p}\). This element is called Osima idempotent associated to \(B\). It aims to play the same role as the symmetry operator in the classical case. Let \(V\) be an \(n\)-dimensional vector space over \(F\). Then as \(S_m\) acts on \(m\)-fold tensor space by place permutation, we have an action of \(G\) on \(\bigotimes^m V\). The space \(V_B(G)=e^*_B(\bigotimes^m V)\) is called the modular symmetry class of tensors associated with \(G\) and \(B\) (each such space is a piece of the block decomposition of \(\bigotimes^m V\) relatively to the \(G\)-action), and the tensor \(e^*_B(v_1\otimes \cdots\otimes v_m)\) is called a modular decomposable symmetrized tensor. The author introduces then the non-degenerate bilinear form \(\langle\;,\;\rangle\) on the modular symmetry class of tensors \(V_B(G)\) which is induced by the usual bilinear form on \(\bigotimes^m V\) defined as \(\langle x_1\otimes\cdots\otimes x_m , y_1\otimes \cdots \otimes y_m\rangle= \prod_{i=1}^m \langle x_i , y_i\rangle\). Using this form, the author investigates sufficient and necessary conditions for vanishing or equality of symmetrized decomposable modular tensors.