A Pieri-type formula and a factorization formula for sums of $K$-$k$-Schur functions [PDF]
, 2018 We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{ \le } g^{(k)}_ $ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by $\widetilde{g}^{(k)}_ $. As an application of this, we also give a $k$-rectangle factorization formula $\widetilde{g}^{(k)}_{R_t\cup }=
Motoki Takigiku
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Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions [PDF]
The Electronic Journal of Combinatorics, 2012 We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam. This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements.
Tom Denton
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Schur function identities, their t-analogs, and k-Schur irreducibility [PDF]
Advances in Mathematics, 2001 18 ...
Luc Lapointe, Jennifer Morse
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A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions [PDF]
International Mathematics Research Notices, 2022 Abstract The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal {F}_{\lambda }^{(k)}$, that generalizes the constructions via the Pieri rule of $K ...
Khanh Nguyen Duc
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Catalan functions and $k$-Schur positivity [PDF]
Journal of the American Mathematical Society, 2018 We prove that gradedkk-Schur functions areGG-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the gradedkk-Schur functions and resolve the Schur positivity andkk-branching conjectures in the strongest possible terms by providing direct ...
Jonah Blasiak +3 more
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A Pieri formula and a factorization formula for sums of
Algebraic Combinatorics, 2019 We give a Pieri-type formula for the sum of K-k-Schur functions ∑ μ≤λ g μ (k) over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, whose sum we denote by g ˜ λ (k) . As an application of this, we also give a k-rectangle factorization formula g ˜ R t ∪λ (k) =g ˜ R t (k) g ˜ λ (k) where R t =(t k+1-t ...
Motoki Takigiku
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A non-commutative generalization of
Discrete Mathematics, 2009 We introduce non-commutative analogues of $k$-Schur functions of Lapointe-Lascoux and Morse. We give an explicit formulas for the expansions of non-commutive functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case.
Nantel Bergeron +2 more
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Lowering operators on $K$-$k$-Schur functions and a lowering operator formula for closed $K$-$k$-Schur functions [PDF]
This paper gives a systematic study of the lowering operators acting on the $K$-$k$-Schur functions, motivated by the pivotal role played by the operators in the definition and study of Katalan functions. A lowering operator formula for closed $K$-$k$-Schur functions is obtained.
Y. Fang, Xing Gao, Li Guo
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Categorification of $k$-Schur functions and refined Macdonald positivity [PDF]
v2, 57 pages. Extensively revised; notation has been updated to align with arXiv:2301.00862, with many additional details and examples. The proof of Theorem 7.1 (prepared through \S5--6) is now more direct. We have also added \S1.1 and expanded \S9--10. Another round of revisions is expected before journal submission.
Syu Kato
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Weighted $K$-$k$-Schur functions and their application to the $K$-$k$-Schur alternating conjecture [PDF]
23 ...
Y. H. Fan, Xing Gao
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