arXiv subjects

Travis Scrimshaw

Publications and source records attributed to Travis Scrimshaw.

At least 37 records · Page 2Linked to original sources

arXiv2022

Crystal invariant theory I: Geometric RSK

Berenstein and Kazhdan's theory of geometric crystals gives rise to two commuting families of geometric crystal operators acting on the space of complex $m \times n$ matrices. These are birational actions, which we view as a crystal-theoretic analogue of the usual action of ${\rm SL}_m \times {\rm SL}_n$ on $m \times n$ matrices. We prove that the field of rational invariants (and ring of polynomial invariants) of each family of geometric crystal operators is generated by a set of algebraically independent polynomials, which are generalizations of the elementary symmetric polynomials in $m$ (or $n$) variables. We also give a set of algebraically independent generators for the intersection of these fields, and we explain how these fields are situated inside the larger fields of geometric $R$-matrix invariants, which were studied by Lam and the third-named author under the name loop symmetric functions. The key tool in our proof is the geometric RSK correspondence of Noumi and Yamada, which we show to be an isomorphism of geometric crystals. In an appendix jointly written with Thomas Lam, we prove the fundamental theorem of loop symmetric functions, which says that the polynomial invariants of the geometric $R$-matrix are generated by the loop elementary symmetric functions.

math.QA↗

arXiv2022

Crystal invariant theory II: Pseudo-energies

The geometric crystal operators and geometric $R$-matrices (or geometric Weyl group actions) give commuting actions on the field of rational functions in $mn$ variables. We study the invariants of various combinations of these actions, which we view as "crystal analogues" of the invariants of $S_m$, ${\rm SL}_m$, $S_n \times S_m$, ${\rm SL}_n \times \, S_m$, and ${\rm SL}_n \times {\rm SL}_m$ acting on the polynomial ring in an $m \times n$ matrix of variables. The polynomial invariants of the $S_m$-action generated by the ${\rm GL}_m$-geometric $R$-matrices were described by Lam and the third-named author as the ring of loop symmetric functions. In a previous paper of the authors, the polynomial invariants of the ${\rm GL}_m$-geometric crystal operators were described as a subring of the ring of loop symmetric functions. In this paper, we give conjectural generating sets for the fields of rational invariants in the remaining cases, and we give formulas expressing a large class of loop symmetric functions in terms of these conjectural generators. Our results include new positive formulas for the central charge and energy function of a product of single-row geometric crystals, and a new derivation of Kirillov and Berenstein's piecewise-linear formula for cocharge. The formulas manifest the symmetries possessed by these functions.

math.QA↗

arXiv2022

Integrable systems and crystals for edge labeled tableaux

We introduce the edge Schur functions $E^λ$ that are defined as a generating series over edge labeled tableaux. We formulate $E^λ$ as the partition function for a solvable lattice model, which we use to show they are symmetric polynomials and derive a Cauchy-type identity with factorial Schur polynomials. Finally, we give a crystal structure on edge labeled tableau to give a positive Schur polynomial expansion of $E^λ$ and show it intertwines with an uncrowding algorithm.

math.CO↗

arXiv2022

K-theoretic crystals for set-valued tableaux of rectangular shapes

In earlier work with C.~Monical, we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials $L_{wλ}$ when $λ$ is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross--Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.

math.CO↗

arXiv2021

Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$

We construct a combinatorial crystal structure on the Kirillov-Reshetikhin crystal $B^{7,s}$ in type $E_7^{(1)}$, where $7$ is the unique node in the orbit of $0$ in the affine Dynkin diagram. We then describe the combinatorial $R$-matrix $R \colon B^{7,s} \otimes B^{7,s'} \to B^{7,s'} \otimes B^{7,s}$.

math.RT↗

arXiv2021

Quasi-solvable lattice models for $\mathrm{Sp}_{2n}$ and $\mathrm{SO}_{2n+1}$ Demazure atoms and characters

We construct colored lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang--Baxter equation that allows us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.

math.CO↗

arXiv2020

Refined dual Grothendieck polynomials, integrability, and the Schur measure

We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grothendieck polynomials conjectured by Grinberg and recover a relation between LPP and the Schur process due to Baik and Rains. Lastly, we extend our vertex model techniques to show some identities for refined Grothendieck polynomials, including a Jacobi-Trudi formula.

math.CO↗

arXiv2020

Double Grothendieck polynomials and colored lattice models

We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt [arXiv:2003.07342] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to given a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials. Taking the stable limit of double Grothendieck polynomials, we obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara. The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.

math.CO↗

arXiv2020

Existence of Kirillov-Reshetikhin crystals for near adjoint nodes in exceptional types

We prove that, in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$, every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion introduced by Kang et.al. In order to apply the criterion, we need to prove some statements concerning values of a bilinear form. We achieve this by using the global bases of extremal weight modules.

math.QA↗

arXiv2020

The regularity of almost-commuting partial Grothendieck--Springer resolutions and parabolic analogs of Calogero--Moser varieties

Consider the moment map $μ\colon T^*(\mathfrak{p} \times \mathbb{C}^n) \to \mathfrak{p}^*$ for a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{gl}_n(\mathbb{C})$. We prove that the preimage of $0$ under $μ$ is a complete intersection when $\mathfrak{p}$ has finitely many $P$-orbits, where $P\subseteq \operatorname{GL}_n(\mathbb{C})$ is a parabolic subgroup such that $\operatorname{Lie}(P) = \mathfrak{p}$, and give an explicit description of the irreducible components. This allows us to study nearby fibers of $μ$ as they are equidimensional, and one may also construct GIT quotients $μ^{-1}(0) /\!\!/_χ P$ by varying the stability condition $χ$. Finally, we study a variety analogous to the scheme studied by Wilson with connections to a Calogero--Moser phase space where only some of particles interact.

math.AG↗

arXiv2020

Uniform description of the rigged configuration bijection

We give a uniform description of the bijection $Φ$ from rigged configurations to tensor products of Kirillov--Reshetikhin crystals of the form $\bigotimes_{i=1}^N B^{r_i,1}$ in dual untwisted types: simply-laced types and types $A_{2n-1}^{(2)}$, $D_{n+1}^{(2)}$, $E_6^{(2)}$, and $D_4^{(3)}$. We give a uniform proof that $Φ$ is a bijection and preserves statistics. We describe $Φ$ uniformly using virtual crystals for all remaining types, but our proofs are type-specific. We also give a uniform proof that $Φ$ is a bijection for $\bigotimes_{i=1}^N B^{r_i,s_i}$ when $r_i$, for all $i$, map to $0$ under an automorphism of the Dynkin diagram. Furthermore, we give a description of the Kirillov--Reshetikhin crystals $B^{r,1}$ using tableaux of a fixed height $k_r$ depending on $r$ in all affine types. Additionally, we are able to describe crystals $B^{r,s}$ using $k_r \times s$ shaped tableaux that are conjecturally the crystal basis for Kirillov--Reshetikhin modules for various nodes $r$.

math.CO↗

arXiv2020

Rigged configurations and the $\ast$-involution for generalized Kac--Moody algebras

We construct a uniform model for highest weight crystals and $B(\infty)$ for generalized Kac--Moody algebras using rigged configurations. We also show an explicit description of the $\ast$-involution on rigged configurations for $B(\infty)$: that the $\ast$-involution interchanges the rigging and the corigging. We do this by giving a recognition theorem for $B(\infty)$ using the $\ast$-involution. As a consequence, we also characterize $B(λ)$ as a subcrystal of $B(\infty)$ using the $\ast$-involution.

math.CO↗

arXiv2020

Colored five-vertex models and Lascoux polynomials and atoms

We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai [arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson [arXiv:1902.01795]. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first known combinatorial interpretation of a Lascoux polynomial and atom. Using this, we prove a conjectured combinatorial interpretation in terms of set-valued tableaux of a Lascoux polynomial and atom due to Pechenik and the second author [arXiv:1904.09674]. We also prove the combinatorial interpretation of the Lascoux atom using set-valued skyline tableaux of Monical [arXiv:1611.08777].

math.CO↗

arXiv2020

Candidate for the crystal $B(-\infty)$ for the queer Lie superalgebra

It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.

math.CO↗

arXiv2019

Multiline queues with spectral parameters

Using the description of multiline queues as functions on words, we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues. We show that the spectral weight of a word is invariant under a natural action of the symmetric group, giving a proof of the commutativity conjecture of Arita, Ayyer, Mallick, and Prolhac. We give a determinant formula for the spectral weight of a word, which gives a proof of a conjecture of the first author and Linusson.

math.CO↗

arXiv2019

Crystal structures for canonical Grothendieck functions

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.

math.CO↗

arXiv2019

On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models

We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a result of Naoi. We use this result to show that, given two tensor products of such KR crystals with the same maximal weight, after removing certain $0$-arrows, the two connected components containing the minimal/maximal elements are isomorphic. Based on the latter fact, we reduce a tensor product of higher level perfect KR crystals to one of single-column KR crystals, which allows us to use the uniform models available in the literature in the latter case. We also use our results to give a combinatorial interpretation of the Q-system relations. Our results are conjectured to extend to the exceptional types.

math.CO↗

arXiv2019

Categorical relations between Langlands dual quantum affine algebras: Exceptional cases

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, $\mathscr{C}_{\mathscr{Q}}^{(1)}$ and $\mathscr{C}_{\mathfrak{Q}}^{(1)}$. These results give Dorey's rule for all exceptional affine types, prove the conjectures of Kashiwara-Kang-Kim and Kashiwara-Oh, and provides the partial answers of Frenkel-Hernandez on Langlands duality for finite dimensional representations of quantum affine algebras of exceptional types.

math.RT↗