Geodesic
A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer
Yuchen Pei1
1Center of Mathematical Sciences and Applications at Harvard University
The electronic journal of combinatorics, Tome 24 (2017) no. 4
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In Matveev-Petrov (2017) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$RSK) 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 $q$RSK, 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 $q$RSK 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 ($q$PNG) and write down the joint distribution of the $q$-polymer partition functions at a fixed time.
DOI : 10.37236/6739
Classification : 05E05, 33D15, 60C05, 60K35
Mots-clés : Robinson-Schensted-Knuth algorithms, Macdonald polynomials, basic hypergeometric series, exactly solvable models

Yuchen Pei  1

1 Center of Mathematical Sciences and Applications at Harvard University 
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     title = {A {\(q\)-Robinson-Schensted-Knuth} algorithm and a \(q\)-polymer},
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Yuchen Pei. A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer. The electronic journal of combinatorics, Tome 24 (2017) no. 4. doi: 10.37236/6739

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