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

TL;DR

This paper investigates two algebraic operators on tensor algebras over a module, analyzing their kernels and relating them to combinatorial shuffle operations, with detailed descriptions depending on the properties of the base ring.

Contribution

It introduces and characterizes the kernels of two tensor algebra operators inspired by the random-to-top shuffle, extending understanding in algebraic and combinatorial contexts.

Findings

Kernel of the alternating operator is a Lie subsuperalgebra.

Kernel of the first operator is described for torsionfree rings.

Descriptions vary when the base ring is over a finite field.

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).