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)
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