The One-Sided Cycle Shuffles in the Symmetric Group Algebra

Explore the mathematical intricacies of a new family of elements in the symmetric group algebra, focusing on the "somewhere-to-below shuffles" in card shuffling. Delve into the properties of these elements, including their near-commutativity and connections to the Fibonacci sequence. Examine the construction of a combinatorial basis for these shuffles and learn how to compute eigenvalues for linear combinations of these elements. Investigate the relationship between these shuffles and other well-known families of elements in the group ring, such as Young-Jucys-Murphy elements. Gain insights into the algebraic structure generated by these shuffles and their applications in card shuffling theory.