The loop Murnaghan–Nakayama rule

Ross, Dustin

doi:10.1007/s10801-013-0436-y

Cited by 8 publications

(12 citation statements)

References 14 publications

(24 reference statements)

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“…Loop Schur functions were introduced by Lam and Pylyavskyy [8] in the context of total positivity of matrix loop groups. References for loop Schur functions are a paper of Lam [7], and more closely related to the topic at hand, a paper of the first author [17].…”

Section: Donaldson-thomas Theory: the Local Picturementioning

confidence: 99%

Cyclic Hodge integrals and loop Schur functions

Ross

Zong

2015

Advances in Mathematics

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We conjecture an evaluation of three-partition cyclic Hodge integrals in terms of loop Schur functions. Our formula implies the orbifold Gromov-Witten/Donaldson-Thomas correspondence for toric Calabi-Yau threefolds with transverse A n singularities. We prove the formula in the case where one of the partitions is empty, and thus establish the orbifold Gromov-Witten/Donaldson-Thomas correspondence for local toric surfaces with transverse A n singularities.

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Section: Donaldson-thomas Theory: the Local Picturementioning

confidence: 99%

Cyclic Hodge integrals and loop Schur functions

Ross

Zong

2015

Advances in Mathematics

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“…The Murnaghan-Nakayama rule gives the Schur expansion of the product of a power sum symmetric function and a Schur function. A loop generalization of this rule was stated in [Lam12], and proved combinatorially by Ross [Ros14]. Here we give a short proof based on Theorem 3.1.…”

Section: The Loop Murnaghan-nakayama Rulementioning

confidence: 73%

A ratio of alternants formula for loop Schur functions

Frieden

2020

Journal of Combinatorics

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Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a "colored weight," and they proved a Jacobi-Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of "loop alternants," extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan-Nakayama rule.other words, the action tropicalizes to a piecewise-linear formula for the combinatorial Rmatrix for tensor products of affine crystals-in this case, the one-row Kirillov-Reshetikhin crystals of type A (1) n−1 [HHI + 01]. This suggests that a function on tensor products of one-row crystals which is invariant under the combinatorial R-matrix should be the tropicalization of a ratio of loop symmetric functions. Lam and Pylyavskyy showed that the intrinsic energy, an important function in affine crystal theory, is in fact the tropicalization of a certain loop Schur function [LP13b] (this loop Schur function turns out to be related to our alternant formulas; see Remark 3.6). Additionally, tensor products of one-row crystals can be viewed as states in the (generalized) Box-Ball system, a well-studied cellular automaton that exhibits soliton behavior [HHI + 01]. Formulas for the scattering of a given state into solitons are conjecturally given by tropicalizations of a cylindric variant of loop Schur functions [LPS].We note that the birational S m action also arose in the context of the local Langlands program [BK00b], and was studied in [Eti03]. Loop symmetric functions have also found application in Gromov-Witten/Donaldson-Thomas theory [RZ13]; this was Ross' motivation for proving the loop Murnaghan-Nakayama rule [Ros14].This paper is organized as follows. In Section 2, we review the basics of loop symmetric functions and the birational S m action. Section 3 contains statements and proofs of the two ratio of alternants formulas (Theorems 3.1 and 3.3), and Section 4 discusses the loop Murnaghan-Nakayama rule (Theorem 4.1).

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“…In our related works [13][14][15][16], we heavily rely on the algebrocombinatorial structure of this formula. For completeness, we reproduce the formula here.…”

Section: Loop Schur Functionsmentioning

confidence: 99%