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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.