Catalan functions and 𝑘-Schur positivity

Blasiak, Jonah; Morse, Jennifer; Anna, Pun,; Summers, Daniel

doi:10.1090/jams/921

Cited by 12 publications

(18 citation statements)

References 35 publications

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“…For instance, Example 4.1 provides the entire column of K −1 indexed by β = [3,1,3]. Specifically, the entries in rows 313, 421, and 520 of this column are +1, the entries in rows 322, 430, and 511 of this column are −1, and all other entries in this column are 0.…”

Section: 5mentioning

confidence: 99%

“…We have v (1) = (3, 0, 2), v (2) = (2, 0, 2), v (3) = (2, 4, 2), and v (4) = (2, 4, 1). We see by inspection that A(v)[i|4] = A(v (i) ) for i = 1, 2, 3, 4.…”

Section: Transitive Tournaments and The Jacobi-trudimentioning

confidence: 99%

“…Remark 5.2. In important recent work [3,4], Blasiak, Morse, Pun, and Summers obtain detailed information about the k-Schur symmetric functions by applying certain products of t-raising operators to Schur functions, creation operators, etc. These products are indexed by root ideals (or their complements) in the poset {(i, j) : 1 i < j m}.…”

Section: T-analoguesmentioning

confidence: 99%

“…Example 5.7. From the tournaments in Example 4.1, we compute all entries in column [3,1,3] of the t-analogue of K −1 : row 313 has 1, row 322 has −t, row 412 has t 2 − t, row 421 has t 2 , row 43 has −t, row 52 has t 2 , row 511 has −t 3 , and all other entries in this column are zero. The tournaments contributing to K −1 (412, 313; t) are the two non-transitive tournaments in T 3 , and we see that the t-powers of these two tournaments are unequal.…”

Section: T-analoguesmentioning

confidence: 99%

“…The following diagram shows four SRHT of shape µ =[3,3,3,3,2].From left to right, these SRHT have total content [42530], [45302], [34511], and [75200]. In more detail, the first SRHT has (a 1 , .…”

mentioning

confidence: 99%

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Combinatorics of the immaculate inverse Kostka matrix

Loehr¹,

Niese²

2022

Algebraic Combinatorics

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The classical Kostka matrix counts semistandard tableaux and expands Schur symmetric functions in terms of monomial symmetric functions. The entries in the inverse Kostka matrix can be computed by various algebraic and combinatorial formulas involving determinants, special rim hook tableaux, raising operators, and tournaments. Our goal here is to develop an analogous combinatorial theory for the inverse of the immaculate Kostka matrix. The immaculate Kostka matrix enumerates dual immaculate tableaux and gives a combinatorial definition of the dual immaculate quasisymmetric functions S * α . We develop several formulas for the entries in the inverse of this matrix based on suitably generalized raising operators, tournaments, and special rim-hook tableaux. Our analysis reveals how the combinatorial conditions defining dual immaculate tableaux arise naturally from algebraic properties of raising operators. We also obtain an elementary combinatorial proof that the definition of S * α via dual immaculate tableaux is equivalent to the definition of the immaculate noncommutative symmetric functions Sα via noncommutative Jacobi-Trudi determinants. A factorization of raising operators leads to bases of NSym interpolating between the S-basis and the h-basis, and bases of QSym interpolating between the S * -basis and the M -basis. We also give t-analogues for most of these results using combinatorial statistics defined on dual immaculate tableaux and tournaments.K n (λ, µ)m µ ; h µ = λ K n (λ, µ)s λ .

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Section: 5mentioning

confidence: 99%

“…We have v (1) = (3, 0, 2), v (2) = (2, 0, 2), v (3) = (2, 4, 2), and v (4) = (2, 4, 1). We see by inspection that A(v)[i|4] = A(v (i) ) for i = 1, 2, 3, 4.…”

Section: Transitive Tournaments and The Jacobi-trudimentioning

confidence: 99%

Section: T-analoguesmentioning

confidence: 99%

Section: T-analoguesmentioning

confidence: 99%

mentioning

confidence: 99%

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Combinatorics of the immaculate inverse Kostka matrix

Loehr¹,

Niese²

2022

Algebraic Combinatorics

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Demazure crystals and the Schur positivity of Catalan functions

Blasiak,

Morse,

Pun

2024

Invent. math.

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Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$ k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_{q}(\widehat{\mathfrak{sl}}_{\ell })$ U q ( sl ˆ ℓ ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

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