Raising operators, recurrences, and the Littlewood–Richardson polynomials

Abstract

The aim of this thesis is to calculate two types of Littlewood–Richardson polynomi- als. These are structure coefficients in the ring of double symmetric functions (x||a) which has a distinguished basis consisting of the double Schur functions sλ(x||a). The first type of Littlewood–Richardson polynomials arises when we consider the product of two double Schur functions, the second when the comultiplication operation in the ring (x||a) is applied to a double Schur function. When the ring (x||a) is specialised to the ring of symmetric functions (x), we recover the Littlewood–Richardson coeffi- cients. Apart from their applications in the combinatorics of symmetric functions, the Littlewood–Richardson polynomials are important for the following reasons. They are applied in geometry and representation theory. The first type of polynomials de- scribe a multiplication rule for equivariant Schubert classes, and also a multiplication rule for virtual quantum immanants and higher Capelli operators. The second type is relevant to describing equivariant cohomology of infinite grassmanians. The structure of this thesis is as follows. In Chapter 1, we introduce well known definitions associated with the ring of symmetric functions (x). Using the Pieri rule and Jacobi–Trudi identity, we then present a proof of a rule used to calculate the Littlewood–Richardson coefficients. This is Theorem 1.8. This proof we present is a simplified version of our main result in Chapter 3. In Chapter 2, we introduce the ring of double symmetric functions (x||a), which is a generalisation of the classical ring (x) depending on an extra set of infinite variables a = (ai)i∈Z . We introduce the basis of double Schur functions, and then explain how the two types of Littlewood–Richardson polynomials arise as structure coefficients involving the double Schur functions. We also discuss the significance of these structure coefficients in combinatorics, representation theory, and geometry. In Chapter 3, we present one of the main results of this thesis using raising operators. This is a new proof of Theorem 3.33, a known formula which calculates the Littlewood–Richardson polynomials arising between the product of two double Schur functions. Our proof relies on two things: first, we introduce a Jacobi–Trudi identity for the double Schur functions. Second, we derive a Pieri rule for the ring (x||a). This Pieri rule is in turn a specialisation of a more general rule which we also introduce for the ring A generated by the indeterminates hr,s from the 9th Variation of Macdonald [14]. In Chapter 4, we discuss the dual Littlewood–Richardson polynomials which arise when comultiplication is applied to the double Schur functions. We also discuss the dual Schur functions and skew double Schur functions. The dual Littlewood– Richardson polynomials then give combinatorial identities involving these functions. In the conclusion of Chapter 4, we present another main result of this thesis. This i is Theorem 4.3, which provides a stable formula to calculate the dual Littlewood– Richardson polynomials. In Chapter 5, we introduce the ring of generalised supersymmetric functions (x/y||a), which has a distinguished basis consisting of generalised Frobenius–Schur functions sλ(x/y||a). Using a recurrence relation, we produce another main result of this thesis. This is a Pieri rule which gives the structure coefficients arising out of the product between the functions sθ(x/y||a) and sλ(x/y||a), where λ is an arbitrary partition and θ is a skew partition not containg a 2 × 2 subdiagram; this is Theo- rem 5.34. A specialisation of this theorem then lets us evaluate some of the dual Littlewood–Richardson coefficients.