A symmetry property for $$q$$ q -weighted Robinson–Schensted and other branching insertion algorithms

Pei, Yuchen

doi:10.1007/s10801-014-0505-x

Cited by 9 publications

(12 citation statements)

References 25 publications

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“…It has been generalised to other types in [12]. A discrete-time version is given in [13] (see also [41]) in the context of Kirillov's geometric RSK mapping on matrices, and a fully discrete q-version, in the context of Ruijenaars q-Toda difference operators and q-Whittaker functions, in [40] (see also [9,43]).…”

Section: Geometric Rsk and Brownian Motionmentioning

confidence: 99%

Geometric RSK and the Toda lattice

O’Connell¹

2013

Illinois J. Math.

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Section: Geometric Rsk and Brownian Motionmentioning

confidence: 99%

Geometric RSK and the Toda lattice

O’Connell¹

2013

Illinois J. Math.

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“…VI]: the Schur functions correspond to q = t, and the Whittaker functions arise in the limit as t = 0 and q 1, [38]. In the recent years, there has been a progress in understanding analogues of the RS correspondences at other levels of the Macdonald hierarchy: q-Whittaker (t = 0 and 0 < q < 1) [59], [65], [16] and Hall-Littlewood (q = 0 and 0 < t < 1) [18]. At these levels, the correspondences become randomized, that is, the image of a deterministic word (as on Fig.…”

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confidence: 99%

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

Матвеев¹,

Petrov²

2016

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q = 0) and geometric (q 1) RSK correspondences (the latter ones are sometimes also called tropical).For 0 < q < 1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q-Whittaker processes (which are t = 0 versions of Macdonald processes of Borodin-Corwin [7]). We present four Markov dynamics which for q = 0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries.Our new two-dimensional discrete time dynamics generalize and extend several known constructions: (1) The discrete time q-TASEPs studied by Borodin-Corwin [8] arise as one-dimensional marginals of our "column" dynamics. In a similar way, our "row" dynamics lead to discrete time q-PushTASEPs -new integrable particle systems in the Kardar-Parisi-Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q-PushASEP conjectured by Corwin-Petrov [23]. (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q-randomized column and row Robinson-Schensted correspondences introduced by O'Connell-Pei [59] and Borodin-Petrov [16], respectively. (3) In a scaling limit as q 1, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma (introduced by Seppäläinen [70]) or strict-weak (introduced independently by O'Connell-Ortmann [58] and Corwin-Seppäläinen-Shen [25]) directed random lattice polymers.

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“…We give the Noumi-Yamada description of this algorithm, from which we obtain a branching growth diagram construction similar to that in [Pei14], and show that the algorithm is symmetric: Theorem 1. Let φA(P, Q) = P(qRSK(A) = (P, Q)) be the probability of obtaining the tableau pair (P, Q) after performing qRSK on matrix A, then…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-Polymer

Pei

2017

Electron. J. Combin.

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In [MP16] a q-deformed Robinson-Schensted-Knuth algorithm (qRSK) was introduced. In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a q-polymer model based on the qRSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and q-geometric weights. We use the q-local moves to define a generalisation of the qRSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the q-polymer in q-geometric environment, formulate a q-version of the multilayer polynuclear growth model (qPNG) and write down the joint distribution of the q-polymer partition functions at a fixed time.arXiv:1610.03692v3 [math.CO]

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A symmetry property for $$q$$ q -weighted Robinson–Schensted and other branching insertion algorithms

Cited by 9 publications

References 25 publications

Geometric RSK and the Toda lattice

Geometric RSK and the Toda lattice

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-Polymer

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