The signed random-to-top operator on tensor space (draft)

Abstract

Given a free module L over a commutative ring k, we study two k-linear operators on the tensor algebra of T(L): One of them sends a pure tensor u_1 (X) u_2 (X) ... (X) u_k to the sum of all tensors u_i (X) u_1 (X) u_2 (X) ... (X) (skip u_i) (X) ... (X) u_k. The other is similar, but the sum is replaced by an alternating sum. These operators can be regarded as algebraic analogues of the "random-to-top shuffle" from combinatorics. We describe the kernel of the second operator (which we call boldface-t); it is a certain easily described Lie subsuperalgebra of T(L). We also describe the kernel of the first operator (which is denoted boldface-t') when the additive group k is torsionfree (the description is analogous to that of the kernel of t) and also when k is an algebra over a finite field (in this case, the description is slightly complicated by the presence of p-th powers).