2 1. INTRODUCTION

the polynomials

˜(k→k

b

)

μλ

(t) can be expressed positively in terms of the branching

polynomials

˜(k)(t)

b

μλ

:=

˜(k−1→k)(t)

b

μλ

,

via iteration (tables of branching polynomials are given in Appendix A).

It has also come to light that ungraded k-Schur functions (the case when t = 1)

are intimately tied to problems in combinatorics, geometry, and representation

theory beyond the theory of Macdonald polynomials. Thus, understanding the

branching coeﬃcients,

˜(k)

b

μλ

:=

˜(k)(1)μλ

b

gives a step-by-step approach to problems in areas such as aﬃne Schubert calculus

and K-theory (see Section 1.4 for more details).

Our work here gives a combinatorial description for the branching coeﬃcients,

proving Conjecture 1.1 when t = 1. We use the ungraded k-Schur functions

sλk)[X](

defined in [14], which coincide with those defined in [9] terms of strong k-tableaux.

Moreover, we conjecture a formula for the branching polynomials in general. The

combinatorics behind these formulas involves a certain poset of k-shapes.

1.2. The poset of k-shapes

A key development in our work is the introduction of a new family of partitions

called k-shapes and a poset on these partitions (see Chapter 2 for full details and

examples). Our formula for the branching coeﬃcients is given in terms of path

enumeration in the poset of k-shapes.

For any partition λ identified by its Ferrers diagram, we define its k-boundary

∂λ to be the cells of λ with hook-length no greater than k. ∂λ is a skew shape, to

which we associate compositions rs(λ) and cs(λ), where rs(λ)i (resp. cs(λ)i) is the

number of cells in the i-th row (resp. column) of ∂λ. A partition λ is said to be a

k-shape if both rs(λ) and cs(λ) are partitions. The rank of k-shape λ is defined to

be |∂λ|, the number of cells in its k-boundary. Πk denotes the set of all k-shapes.

We introduce a poset structure on Πk where the partial order is generated by

distinguished downward relations in the poset called moves (Definition 2.14). The

set of k-shapes contains the set

Ck

of all k-cores (partitions with no cells of hook-

length k) and the set

Ck+1

of k + 1-cores. Moreover, the maximal elements of

Πk

are given by

Ck+1

and the minimal elements by

Ck.

In Definition 3.1 we give a

charge statistic on moves from which we obtain an equivalence relation on paths

(sequences of moves) in

Πk;

roughly speaking, two paths are equivalent if they are

related by a sequence of charge-preserving diamonds (see Eqs. (3.1)-(3.3)). Charge

is thus constant on equivalence classes of paths.

For λ, μ ∈

Πk, Pk(λ,

μ) is the set of paths in

Πk

from λ to μ and

Pk(λ,

μ) is

the set of equivalence classes in

Pk(λ,

μ). Our main result is that the branching

coeﬃcients enumerate these equivalence classes. To be precise, for λ ∈

Ck+1

and

μ ∈

Ck,

set

bμλ

(k)

(t) :=

˜(k)

b

rs(μ)rs(λ)

(t) (1.5)

bμλ

(k)

:=

˜(k)

b

rs(μ)rs(λ)

(1.6)