Multiline Queues with Spectral Parameters

Aas, Erik; Grinberg, Darij; Scrimshaw, Travis

doi:10.1007/s00220-020-03694-4

Cited by 7 publications

(11 citation statements)

References 65 publications

(81 reference statements)

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“…And in the case that all particles have types 0, 1, 2, nonintersecting paths were used to give explicit determinantal formulas for steady state probabilities in [Man17, Theorem 2.6]. Multiline queues were also connected to tableaux via nonintersecting paths in [AGS20], though the weights on queues there were different from ours. Definition 8.1.…”

Section: Multiline Queues and Steady State Probabilitiesmentioning

confidence: 99%

Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

Kim¹,

Williams²

2021

Preprint

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Consider a lattice of n sites arranged around a ring, with the n sites occupied by particles of weights {1, 2, . . . , n}; the possible arrangements of particles in sites thus corresponds to the n! permutations in S n . The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on S n , in which two adjacent particles of weights i < j swap places at rate x i − y n+1−j if the particle of weight j is to the right of the particle of weight i. (Otherwise nothing happens.) When y i = 0 for all i, the stationary distribution was conjecturally linked to Schubert polynomials [LW12], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [AL14, AM13]. In the case of general y i , Cantini [Can16] showed that n of the n! states have probabilities proportional to double Schubert polynomials. In this paper we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns 2413, 4132, 4213 and 3214. We show that there areevil-avoiding permutations in S n , and for each evil-avoiding permutation w, we give an explicit formula for the steady state probability ψ w as a product of double Schubert polynomials. (Conjecturally all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When y i = 0 for all i, we give multiline queue formulas for the z-deformed steady state probabilities, and use this to prove the monomial factor conjecture from [LW12]. Finally, we show that the Schubert polynomials that arise in our formulas are flagged Schur functions, and give a bijection in this case between multiline queues and semistandard Young tableaux.

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Section: Multiline Queues and Steady State Probabilitiesmentioning

confidence: 99%

Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

Kim¹,

Williams²

2021

Preprint

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“…Let λ, µ ∈ Par n , r, s ∈ P n with µ ⊆ λ and let 1 ≤ k ≤ n − 1 be an integer such that µ k ≥ λ k+1 . If φ β is the map defined by shifting 1) ,s (1) ) λ (1) /µ (1) 2) ,s (2) ) λ (2) /µ (2) (x; α, β) , 1) ,s (1) ) 2) ,s (2) )…”

Section: Flagged Grothendieck Polynomialsmentioning

confidence: 99%

“…Let n > 1 and suppose that the statement holds for all integers less than n. If µ t ≥ λ t+1 for some 1 ≤ t ≤ n − 1, then we set γ (1) = (γ 1 , . .…”

Section: Flagged Grothendieck Polynomialsmentioning

confidence: 99%

“…Let λ, µ ∈ Par n , r, s ∈ P n with µ ⊆ λ and suppose 1 ≤ k ≤ n − 1 is an integer such that µ k ≥ λ k+1 . If φ α is the map defined by shifting α i to α i+1 , then 1) ,s (1) )…”

Section: Flagged Grothendieck Polynomialsmentioning

confidence: 99%

“…We also note that dented partitions are a special case of pseudo-partitions introduced by Aas, Grinberg, and Scrimshaw, who found a Jacobi-Trudi-type formula for the generating function for semistandard Young tableaux of a pseudo-partition shape with a flag condition in [1,Theorem 5.3]. Definition 7.12.…”

Section: Flagged Dual Grothendieck Polynomialsmentioning

confidence: 99%

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Refined canonical stable Grothendieck polynomials and their duals

Hwang,

Jang,

Kim

et al. 2021

Preprint

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In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi-Trudi-type formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials. We also consider flagged versions of Grothendieck polynomials and their duals with skew shapes.

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Quasi-solvable lattice models for and Demazure atoms and characters

Buciumas

Scrimshaw

2022

Forum of Mathematics, Sigma

Self Cite

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We construct coloured lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang–Baxter equation that allow us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.

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Multiline Queues with Spectral Parameters

Cited by 7 publications

References 65 publications

Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations

Refined canonical stable Grothendieck polynomials and their duals

Quasi-solvable lattice models for and Demazure atoms and characters

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