q-Whittaker functions, finite fields, and Jordan forms
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| Format: | Preprint |
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2022
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| _version_ | 1866912224641024000 |
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| author | Karp, Steven N. Thomas, Hugh |
| author_facet | Karp, Steven N. Thomas, Hugh |
| contents | The $q$-Whittaker function $W_λ(\mathbf{x};q)$ associated to a partition $λ$ is a $q$-analogue of the Schur function $s_λ(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial $P_λ(\mathbf{x};q,t)$. We show combinatorially how to expand $W_λ(\mathbf{x};q)$ in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size $1/q$. This yields an expression analogous to a well-known formula for the Hall-Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, proving the Cauchy identity for $q$-Whittaker functions. We call our probabilistic bijection the $q$-Burge correspondence, and prove that in the limit as $q\to 0$, we recover a description of the classical Burge correspondence (also known as column RSK) due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary double coset of $\text{GL}_n$ modulo two parabolic subgroups, which we find to be of independent interest. As an application, we use the $q$-Burge correspondence to count isomorphism classes of certain modules over the preprojective algebra of a type $A$ quiver (i.e. a path), refined according to their socle filtrations. This develops a connection between the combinatorics of symmetric functions and the representation theory of preprojective algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_12590 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | q-Whittaker functions, finite fields, and Jordan forms Karp, Steven N. Thomas, Hugh Combinatorics Probability Representation Theory 05E05, 60C05, 14M15, 15B33, 16G20, 05A30 The $q$-Whittaker function $W_λ(\mathbf{x};q)$ associated to a partition $λ$ is a $q$-analogue of the Schur function $s_λ(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial $P_λ(\mathbf{x};q,t)$. We show combinatorially how to expand $W_λ(\mathbf{x};q)$ in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size $1/q$. This yields an expression analogous to a well-known formula for the Hall-Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, proving the Cauchy identity for $q$-Whittaker functions. We call our probabilistic bijection the $q$-Burge correspondence, and prove that in the limit as $q\to 0$, we recover a description of the classical Burge correspondence (also known as column RSK) due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary double coset of $\text{GL}_n$ modulo two parabolic subgroups, which we find to be of independent interest. As an application, we use the $q$-Burge correspondence to count isomorphism classes of certain modules over the preprojective algebra of a type $A$ quiver (i.e. a path), refined according to their socle filtrations. This develops a connection between the combinatorics of symmetric functions and the representation theory of preprojective algebras. |
| title | q-Whittaker functions, finite fields, and Jordan forms |
| topic | Combinatorics Probability Representation Theory 05E05, 60C05, 14M15, 15B33, 16G20, 05A30 |
| url | https://arxiv.org/abs/2207.12590 |