$q$-Whittaker polynomials: bases, branching and direct limits
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| Format: | Preprint |
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2024
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| _version_ | 1866929609306537984 |
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| author | Bhattacharya, Aritra Ratheesh, T V Viswanath, Sankaran |
| author_facet | Bhattacharya, Aritra Ratheesh, T V Viswanath, Sankaran |
| contents | We study $q$-Whittaker polynomials and their monomial expansions given by the fermionic formula, the inv statistic of Haglund-Haiman-Loehr and the quinv statistic of Ayyer-Mandelshtam-Martin. The combinatorial models underlying these expansions are partition overlaid patterns and column strict fillings. The former model is closely tied to representations of the affine Lie algebra $\widehat{\mathfrak{sl}_n}$ and admits projections, branching maps and direct limits that mirror these structures in the Chari-Loktev basis of local Weyl modules. We formulate novel versions of these notions in the column strict fillings model and establish their main properties. We construct weight-preserving bijections between the models which are compatible with projection, branching and direct limits. We also establish connections to the coloured lattice paths formalism for $q$-Whittaker polynomials due to Wheeler and collaborators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $q$-Whittaker polynomials: bases, branching and direct limits Bhattacharya, Aritra Ratheesh, T V Viswanath, Sankaran Combinatorics Representation Theory 05E10, 05E05 We study $q$-Whittaker polynomials and their monomial expansions given by the fermionic formula, the inv statistic of Haglund-Haiman-Loehr and the quinv statistic of Ayyer-Mandelshtam-Martin. The combinatorial models underlying these expansions are partition overlaid patterns and column strict fillings. The former model is closely tied to representations of the affine Lie algebra $\widehat{\mathfrak{sl}_n}$ and admits projections, branching maps and direct limits that mirror these structures in the Chari-Loktev basis of local Weyl modules. We formulate novel versions of these notions in the column strict fillings model and establish their main properties. We construct weight-preserving bijections between the models which are compatible with projection, branching and direct limits. We also establish connections to the coloured lattice paths formalism for $q$-Whittaker polynomials due to Wheeler and collaborators. |
| title | $q$-Whittaker polynomials: bases, branching and direct limits |
| topic | Combinatorics Representation Theory 05E10, 05E05 |
| url | https://arxiv.org/abs/2412.00116 |