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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2408.02040 |
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| _version_ | 1866917742734475264 |
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| author | Gaetz, Christian Goldin, Rebecca Knutson, Allen |
| author_facet | Gaetz, Christian Goldin, Rebecca Knutson, Allen |
| contents | In [Hamaker-Pechenik-Speyer-Weigandt, Nenashev, Pechenik-Weigandt] are studied certain operators on polynomials and power series that commute with all divided difference operators $\partial_i$. We introduce a second set of "martial" operators {\martial_i} that generate the full commutant, and show how a Hopf-algebraic approach naturally reproduces the operators $ξ^ν$ from [Nenashev]. We then pause to study Klyachko's homomorphism $H^*(Fl(n)) \to H^*($the permutahedral toric variety$)$, and extract the part of it relevant to Schubert calculus, the "affine-linear genus''. This genus is then re-obtained using Leibniz combinations of the {\martial_i}. We use Nadeau-Tewari's $q$-analogue of Klyachko's genus to study the equidistribution of $\ell$ and comaj on $[n]\choose k$, generalizing known results on $S_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The commutant of divided difference operators, Klyachko's genus, and the comaj statistic Gaetz, Christian Goldin, Rebecca Knutson, Allen Combinatorics 05E14 20C08 In [Hamaker-Pechenik-Speyer-Weigandt, Nenashev, Pechenik-Weigandt] are studied certain operators on polynomials and power series that commute with all divided difference operators $\partial_i$. We introduce a second set of "martial" operators {\martial_i} that generate the full commutant, and show how a Hopf-algebraic approach naturally reproduces the operators $ξ^ν$ from [Nenashev]. We then pause to study Klyachko's homomorphism $H^*(Fl(n)) \to H^*($the permutahedral toric variety$)$, and extract the part of it relevant to Schubert calculus, the "affine-linear genus''. This genus is then re-obtained using Leibniz combinations of the {\martial_i}. We use Nadeau-Tewari's $q$-analogue of Klyachko's genus to study the equidistribution of $\ell$ and comaj on $[n]\choose k$, generalizing known results on $S_n$. |
| title | The commutant of divided difference operators, Klyachko's genus, and the comaj statistic |
| topic | Combinatorics 05E14 20C08 |
| url | https://arxiv.org/abs/2408.02040 |