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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.16641 |
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| _version_ | 1866910888107180032 |
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| author | Arroyo, Joshua Hamaker, Zachary Hawkes, Graham Pan, Jianping |
| author_facet | Arroyo, Joshua Hamaker, Zachary Hawkes, Graham Pan, Jianping |
| contents | We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur $Q$-functions, which are indexed by strict partitions. A combinatorial description of the Schur $Q$- coefficients is given by Kraśkiewicz insertion. Similarly, their $K$-Stanley analogues are conjectured to expand positively into $GQ$'s, which are $K$-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse also indexed by strict partitions. We introduce a $K$-theoretic analogue of Kraśkiewicz insertion, which can be used to enumerate 0-Hecke expressions for signed permutations and gives a conjectural combinatorial rule for computing this $GQ$ expansion.
We show the Type C $K$-Stanleys for certain fully commutative signed permutations are skew $GQ$'s. Combined with a Pfaffian formula of Anderson's, this allows us to prove Lewis and Marberg's conjecture that $GQ$'s of (skew) rectangle shape are $GQ$'s of trapezoid shape. Combined with our previous conjecture, this also gives an explicit combinatorial description of the skew $GQ$ expansion into $GQ$'s. As a consequence, we obtain a conjecture for the product of two $GQ$ functions where one has trapezoid shape. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Type C $K$-Stanley symmetric functions and Kraśkiewicz-Hecke insertion Arroyo, Joshua Hamaker, Zachary Hawkes, Graham Pan, Jianping Combinatorics 05E05 We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur $Q$-functions, which are indexed by strict partitions. A combinatorial description of the Schur $Q$- coefficients is given by Kraśkiewicz insertion. Similarly, their $K$-Stanley analogues are conjectured to expand positively into $GQ$'s, which are $K$-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse also indexed by strict partitions. We introduce a $K$-theoretic analogue of Kraśkiewicz insertion, which can be used to enumerate 0-Hecke expressions for signed permutations and gives a conjectural combinatorial rule for computing this $GQ$ expansion. We show the Type C $K$-Stanleys for certain fully commutative signed permutations are skew $GQ$'s. Combined with a Pfaffian formula of Anderson's, this allows us to prove Lewis and Marberg's conjecture that $GQ$'s of (skew) rectangle shape are $GQ$'s of trapezoid shape. Combined with our previous conjecture, this also gives an explicit combinatorial description of the skew $GQ$ expansion into $GQ$'s. As a consequence, we obtain a conjecture for the product of two $GQ$ functions where one has trapezoid shape. |
| title | Type C $K$-Stanley symmetric functions and Kraśkiewicz-Hecke insertion |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2503.16641 |