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We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus-g, n-marked curve C to Pr, sending the marked points on C to specified general points in Pr, is equal to (r + 1)g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a combinatorial interpretation in terms of Young tableaux by the Littlewood-Richardson rule. We give a bijection, generalizing the RSK correspondence, between the tableaux in question and the (r+1)ary sequences of length g, and we explore our bijection’s combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which r = 1 and several marked points map to the same point in P1, the number of morphisms is still 2g for sufficiently large d.
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Combinatorics of Generalized Exponents
Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $$A_{n-1}$$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $$C_{n}$$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $$A_{2n-1}$$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $$t$$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $$C$$ branching rules. Our methods are expected to extend to the orthogonal types.
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- PAR ID:
- 10186619
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2020
- Issue:
- 16
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 4942 to 4992
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rny157
