The shape of some linear transformations

Recently, Ito and Terwilliger proposed a problem about some linear transformations on a finite dimensional vector space V over a field F, which can be seen as a generalization of the notions of tridiagonal pair and q-inverting pair (see Problem 1.2). In this paper, we will introduce the notion of the shape of the linear transformations . Our main results are the properties of the shape: (i) The shape is symmetric and unimodal; (ii) Let σ denote an anti-isomorphism from End(V) to End(V′) with V′ a vector space over F with the same dimension as V. Assume that the linear transformations on V satisfy all conditions of Problem 1.2. Then the linear transformations also satisfy all conditions of Problem 1.2. Moreover, have the same shape as .