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) sλ ,

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] = sλ 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