Abstract: Alternating sign matrices (ASMs) are square matrices with entries in {−1, 0, 1}, where in every row and column, the non-zero entries alternate in sign and sum to 1. They were introduced by Robbins and Rumsey in the 1980s in connection with their work on λ-determinants (a family of generalisations of the usual determinant), which can be written as a sum over all ASMs of a given size. Guided by numerical data, they proposed that the number of n × n-ASMs is given by a remarkably simple product formula.
The resulting sequence 1, 2, 7, 42, 429, 7436, 218348, . . . surfaces in the enumeration of many combinatorial families: some admit straightforward bijections with ASMs, while for others no such correspondence is known to date. Although satisfactory bijective proofs are lacking in most cases, many of these results have been established algebraically via lengthy and intricate computations.
In recent years, the modified Robbins polynomials and various Littlewood identities for them have played a key role in many proofs of results of this type or refinements thereof. This thesis is devoted to a study of the modified Robbins polynomials from the perspective of symmetric function theory.
We develop analogues of various classical results, such as the Cauchy identity and Littlewood identities, and provide an expansion of the modified Robbins polynomials in the Schur basis of the ring of symmetric polynomials. We study two combinatorial models for the modified Robbins polynomials and use them to establish bijective proofs for various identities relating the modified Robbins polynomials to other well-known symmetric polynomials like Schur polynomials or the Grassmann-Grothendieck polynomials.
Additionally, we provide a bijection between the underlying combinatorial models of the modified Robbins polynomials and a special case of the fully inhomogeneous spin Hall-Littlewood symmetric rational functions. Guided by this close connection, reminiscent of the correspondence between ASMs and the six-vertex model, we prove two new Littlewood identities for the fully inhomogeneous spin Hall-Littlewood symmetric rational functions, generalising the classical Littlewood identity.
As an application of a bounded Littlewood identity for the (modified) Robbins polynomials, we establish an equidistribution of two statistics on alternating sign pentagons and Magog pentagons and discuss the applications and implications of this result.