Haglund’s conjecture on 3-column Macdonald polynomials

Blasiak, Jonah

doi:10.1007/s00209-015-1612-7

Cited by 14 publications

(54 citation statements)

References 21 publications

(89 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients g λµν where one of the partitions is a hook, recovering previous results of the two authors [5,21]. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux [19].• reprove and strengthen the rule from [21],• establish a precise connection between this rule and the Lascoux heuristic, and • uncover a surprising parallel between this rule and combinatorics underlying transformed Macdonald polynomials indexed by a 3-column shape, as described in [14,6].…”

supporting

confidence: 72%

“…In [6], the first author used the theory of noncommutative Schur functions to prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by 3-tuples of skew shapes and thereby settled a conjecture of Haglund [14] on transformed Macdonald polynomials indexed by 3-column shapes. The noncommutative Schur function computation required for this work ([6, Theorem 1.1]) is quite similar to Theorem 2.3 (though it seems neither result can be obtained from the other; see §7.2).…”

Section: Resultsmentioning

confidence: 99%

“…• establish a precise connection between this rule and the Lascoux heuristic, and • uncover a surprising parallel between this rule and combinatorics underlying transformed Macdonald polynomials indexed by a 3-column shape, as described in [14,6].…”

mentioning

confidence: 84%

“…Section 5 develops tableau combinatorics for the proof of the main theorem, and Section 6 is devoted to its proof. In Section 7, we conjecture a strengthening of the main theorem and compare it to results about LLT polynomials from [6]. Finally, Section 8 lays out the basic theory of noncommutative super Schur functions, including two results about when elementary functions commute.…”

Section: 7mentioning

confidence: 95%

“…We develop a theory of noncommutative super Schur functions based on work of Fomin-Greene [11] and the first author [6]. Using this we…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Kronecker coefficients and noncommutative super Schur functions

Blasiak

Liu

2018

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene [11]. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients g λµν where one of the partitions is a hook, recovering previous results of the two authors [5,21]. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux [19].• reprove and strengthen the rule from [21],• establish a precise connection between this rule and the Lascoux heuristic, and • uncover a surprising parallel between this rule and combinatorics underlying transformed Macdonald polynomials indexed by a 3-column shape, as described in [14,6].

show abstract

supporting

confidence: 72%

Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 84%

Section: 7mentioning

confidence: 95%

“…We develop a theory of noncommutative super Schur functions based on work of Fomin-Greene [11] and the first author [6]. Using this we…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Kronecker coefficients and noncommutative super Schur functions

Blasiak

Liu

2018

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

show abstract

Noncommutative Schur functions, switchboards, and Schur positivity

Blasiak

Fomin

2016

Sel. Math. New Ser.

Self Cite

View full text Add to dashboard Cite

show abstract

Modified Macdonald Polynomials and Integrability

Garbali

Wheeler

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

We derive combinatorial formulae for the modified Macdonald polynomial H λ (x; q, t) using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to the quantum group of Uq( sln+1). and where the summation in (4) runs over flags of partitions {ν} = {∅ ≡ ν 0 ⊆ ν 1 ⊆ · · · ⊆ ν N ≡ λ ′ } such that |ν k | = µ 1 + · · · + µ k for all 1 k N . Here we have used the standard definitions [37] of partition conjugate λ ′ and weight |ν|.One of the preliminary results of the present work is an explanation of the formula (4) at the level of exactly solvable lattice models; this is given in Section 3. It is natural to expect such an interaction with integrability: connections of the (ordinary) Hall-Littlewood polynomials P λ (x; t) with the integrable t-boson model have been known for some time [44,45,26], and this theory was extended to finite-spin lattice models by Borodin in [6], yielding a rational one-parameter deformation of the Hall-Littlewood family.1.2. Macdonald polynomials and expansions. Both of the coefficients K ν,λ (t) and P λ,µ (t) can be naturally generalized to the Macdonald level. The two-parameter Kostka-Foulkes polynomials K ν,λ (q, t) are defined by the expansionwhere H λ (x; q, t) denotes a modified Macdonald polynomial; the latter will be defined in Section 2.In [36,37], Macdonald conjectured that K ν,λ (q, t) ∈ N[q, t]; this problem, the positivity conjecture, developed great notoriety and was only fully resolved more than ten years later by Haiman [19]. The equation (6) has an interpretation in terms of the Hilbert scheme of N points [17] and indeed the proof of [19] relied heavily on geometric techniques, which did not yield a direct formula for K ν,λ (q, t). It is a major outstanding problem to find a manifestly positive combinatorial expression for these coefficients, although this has been solved in some partial cases [21,43,1,2]. Similarly to above, one can convert (6) to an expansion over the monomial symmetric functions, leading to two-parameter coefficients P λ,µ (q, t):Combinatorially speaking, more is known about P λ,µ (q, t) than K ν,λ (q, t). In particular, a manifestly positive expression for P λ,µ (q, t), in terms of fillings of tableaux with certain associated statistics, was obtained by Haglund-Haiman-Loehr in [21].The current paper will also be concerned with the study of P λ,µ (q, t). We offer a new, positive formula for these coefficients, which is of a very different nature to the formula obtained in [21]; rather, it is in the same spirit as Kirillov's expansion (4), and can be seen to degenerate to the latter at q = 0. Let us now start to describe it.

show abstract

Haglund’s conjecture on 3-column Macdonald polynomials

Cited by 14 publications

References 21 publications

Kronecker coefficients and noncommutative super Schur functions

Kronecker coefficients and noncommutative super Schur functions

Noncommutative Schur functions, switchboards, and Schur positivity

Modified Macdonald Polynomials and Integrability

Contact Info

Product

Resources

About