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Crystal structures for canonical Grothendieck functions

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Graham Hawkes, Travis Scrimshaw

Combinatorics · Submitted 26 Jul 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known.

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arXiv ID1907.11415 (version 2)
Pith IDJE4N32HT, Pith's permanent record of this paper and its review
CitationGraham Hawkes, Travis Scrimshaw. "Crystal structures for canonical Grothendieck functions." arXiv:1907.11415 (2019). https://pith.science/paper/JE4N32HT
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What the abstract itself states, as probabilities: claims first 0.21 · states limitation 0.46 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.59 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also give an extension of Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. contribution 0.96 0.96 0.54 asserted 1.00
s1 We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. result 0.81 0.96 0.15 asserted 1.00
s2 As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. interpretation 0.51 0.91 0.54 asserted 1.00
s3 We also give an extension of Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions. contribution 0.95 0.94 0.07 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also give an extension of Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions.

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