Flow Polytopes and the Space of Diagonal Harmonics

Liu, Ricky Ini; Morales, Alejandro H.; Mészáros, Karola

doi:10.4153/cjm-2018-007-3

Cited by 7 publications

(6 citation statements)

References 28 publications

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“…. , α n ) of n positive integers, an α-Tesler matrix U = (u i,j ) 1≤i,j≤n is an n × n upper triangular matrix with nonnegative integer entries such that, for i = 1 to n, This corollary follows from work in [1,9,14]. It would be interesting to find an extension of this corollary to the entire graded Frobenius series of R n,k,r for general r.…”

Section: 2mentioning

confidence: 99%

Tail Positive Words and Generalized Coinvariant Algebras

Rhoades

Wilson

2017

Electron. J. Combin.

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Let n, k, and r be nonnegative integers and let Sn be the symmetric group. We introduce a quotient R n,k,r of the polynomial ring Q[x1, . . . , xn] in n variables which carries the structure of a graded Sn-module. When r ≥ n or k = 0 the quotient R n,k,r reduces to the classical coinvariant algebra Rn attached to the symmetric group. Just as algebraic properties of Rn are controlled by combinatorial properties of permutations in Sn, the algebra of R n,k,r is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of R n,k,r and its graded Sn-isomorphism type. We also view R n,k,r as a module over the 0-Hecke algebra Hn(0), prove that R n,k,r is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient R n,k,r and the delta operators of the theory of Macdonald polynomials.

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Section: 2mentioning

confidence: 99%

Tail Positive Words and Generalized Coinvariant Algebras

Rhoades

Wilson

2017

Electron. J. Combin.

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“…A prominent example of this is the study of flows on graphs and flow polytopes associated to transportation networks, which have been the subject of intense study in recent years. Given a directed acyclic graph (DAG) G with capacity one on every edge, the polytope of flows of strength one is a lattice polytope with vertices corresponding to maximal routes in G. In this paper, we call this the flow polytope for G. Flow polytopes are a central object of study in combinatorial optimization, and they also have important connections with various areas including representation theory [5], diagonal harmonics [24], Grothendieck polynomials [23,26], and toric geometry [19].…”

Section: Introductionmentioning

confidence: 99%

Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras

Bell¹,

Braun²,

Bruegge³

et al. 2022

Preprint

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The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and τ -tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart h * -polynomials.

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“…The flow polytope F G associated to a directed acyclic graph G is the set of all flows f : E(G) → R 0 of size one. Flow polytopes are fundamental objects in combinatorial optimization [17], and in the past decade they were also uncovered in representation theory [1,12], the study of the space of diagonal harmonics [8,13], and the study of Schubert and Grothendieck polynomials [4,5]. A natural way to analyze a convex polytope is to dissect it into simplices.…”

Section: Introductionmentioning

confidence: 99%

“…For a simple graph G, F ⊆ E(G\0), and {z F I } the parameters defined by(8), Newton(L G,F (t)) is the generalized permutahedronNewton(L G,F (t)) = Conv(LD(G, F )) = P z n {z F I } I⊆[n] . Furthermore, each integer point of P z n {z F I } is in LD(G, F ), so L G,F (t) has SNP.Algebraic Combinatorics, Vol.…”

mentioning

confidence: 99%

From generalized permutahedra to Grothendieck polynomials via flow polytopes

Mészáros¹,

Dizier²

2020

Algebraic Combinatorics

Self Cite

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We study a family of dissections of flow polytopes arising from the subdivision algebra. To each dissection of a flow polytope, we associate a polynomial, called the left-degree polynomial, which we show is invariant of the dissection considered (proven independently by Grinberg). We prove that left-degree polynomials encode integer points of generalized permutahedra. Using that certain left-degree polynomials are related to Grothendieck polynomials, we resolve special cases of conjectures by Monical, Tokcan, and Yong regarding the saturated Newton polytope property of Grothendieck polynomials.

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Flow Polytopes and the Space of Diagonal Harmonics

Cited by 7 publications

References 28 publications

Tail Positive Words and Generalized Coinvariant Algebras

Tail Positive Words and Generalized Coinvariant Algebras

Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras

From generalized permutahedra to Grothendieck polynomials via flow polytopes

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