CHAPTER 1
Introduction
1.1. k-Schur functions and branching coefficients
The theory of k-Schur functions arose from the study of Macdonald polynomials
and has since been connected to quantum and affine Schubert calculus, K-theory,
and representation theory [2, 3, 4, 7, 10, 15]. The origin of the k-Schur functions
is related to Macdonald’s positivity conjecture, which asserted that in the expansion
(1.1)
Hμ[X; q, t] =
λ
Kλμ(q, t) ,
the coefficients Kλμ(q, t), called q, t-Kostka polynomials, belong to Z≥0[q, t]. Al-
though the final piece in the proof of this conjecture was made by Haiman [5] using
representation theoretic and geometric methods, the long study of this conjecture
brought forth many further problems and theories. The study of the q, t-Kostka
polynomials remains a matter of great interest.
It was conjectured in [11] that by fixing an integer k 0, any Macdonald
polynomial indexed by λ
Bk
(the set of partitions such that λ1 k) could be
decomposed as:
(1.2) Hμ[X; q, t ] =
λ∈Bk
Kλμ)(q, (k
t)
sλk)[X; (
t ] where
Kλμ)(q, (k
t) Z≥0[q, t] ,
for some symmetric functions
sλk)[X; (
t] associated to sets of tableaux called atoms.
Conjecturally equivalent characterizations of
sλk)[X; (
t] were later given in [12, 9]
and the descriptions of [11, 12, 9] are now all generically called (graded) k-Schur
functions. A basic property of the k-Schur functions is that
(1.3)
sλk)[X; (
t] = for k |λ| ,
and it thus follows that Eq. (1.2) significantly refines Macdonald’s original conjec-
ture since the expansion coefficient
Kλμ)(q, (k
t) reduces to Kλμ(q, t) for large k.
Furthermore, it was conjectured that the k-Schur functions satisfy a highly
structured filtration, which is our primary focus here. To be precise:
Conjecture 1.1. For k k and partitions μ
Bk
and λ
Bk
, there are
polynomials
˜(k→k
b
)
μλ
(t) Z≥0[t] such that
(1.4)
sμk)[X; (
t] =
λ∈Bk
˜(k→k
b
)
μλ
(t)
sλk ( )
[X; t].
In particular, the Schur function expansion of a k-Schur function is obtained from
(1.3) and (1.4) by letting k ∞. The remarkable property described in Conjec-
ture 1.1 provides a step-by-step approach to understanding k-Schur functions since
1
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