Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916872618770432 |
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| author | Fan, Yaozhou Gao, Xing |
| author_facet | Fan, Yaozhou Gao, Xing |
| contents | We introduce the new concept of weighted $K$-$k$-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both $K$-$k$-Schur functions and closed $k$-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed $k$-bounded partitions. As a central application, we resolve the $K$-$k$-Schur alternating conjecture -- posed by Blasiak, Morse, and Seelinger in 2022 -- for a wide class of $k$-bounded partitions, including all strictly decreasing $k$-bounded partitions. Our results shed new light on the combinatorial structure of $K$-theoretic symmetric functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture Fan, Yaozhou Gao, Xing Combinatorics 05E05, 05E10, 14N15 We introduce the new concept of weighted $K$-$k$-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both $K$-$k$-Schur functions and closed $k$-Schur Katalan functions. This new notion exhibits a fundamental alternating property under certain conditions on the indexed $k$-bounded partitions. As a central application, we resolve the $K$-$k$-Schur alternating conjecture -- posed by Blasiak, Morse, and Seelinger in 2022 -- for a wide class of $k$-bounded partitions, including all strictly decreasing $k$-bounded partitions. Our results shed new light on the combinatorial structure of $K$-theoretic symmetric functions. |
| title | Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture |
| topic | Combinatorics 05E05, 05E10, 14N15 |
| url | https://arxiv.org/abs/2507.23222 |