1,720,977 research outputs found
New identities for theta operators
Full text linkIn this article, we prove a new general identity involving the Theta operators introduced by the first author, Iraci, and Vanden Wyngaerd [Adv. Math. 376 (2021), p.59]. From this result, we can easily deduce several new identities that have combinatorial consequences in the study of Macdonald polynomials and diagonal coinvariants. In particular, we provide a unifying framework from which we recover many identities scattered in the literature, often resulting in drastically shorter proofs
Entropy and Følner function in algebras
No full textAbstractWe introduce the notion of entropic Følner function for algebras and we study its relation with the isoperimetric profile and the lower transcendence degree.Under the assumption of a technical conjecture we use the Shannon inequality to derive a theorem on the lower transcendence degree of domains and division algebras. We finally discuss its relation with some old conjectures of M. Artin, L. Small and J. Zhang
-Positivity of vertical strip LLT polynomials
No full textIn this article we prove the e-positivity of Gν[X;q+1] when Gν[X;q] is a vertical strip LLT polynomial. This property has been conjectured in [2] and [7], and it implies several e-positivities conjectured in those references and in [3]. We make use of a result of Carlsson and Mellit [5] that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of the Dyck path algebra to the constant 1. Our proof gives in fact an algorithm to expand these symmetric functions in the elementary basis, and it shows, as a byproduct, that these compositions of operators are actually multiplication operators.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Arithmetic matroids and Tutte polynomials
Full text linkWe introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula
A proof of the compositional Delta conjecture
Full text linkWe prove a compositional refinement of the Delta conjecture (rise version) of Haglund, Remmel and Wilson [16] for Δejavax.xml.bind.JAXBElement@350ecd2e′en which was stated in [8] in terms of Theta operators
Arithmetic matroids, the Tutte polynomial and toric arrangements
No full textWe introduce the notion of an arithmetic matroid whose main example is a list of elements of a finitely generated abelian group. In particular, we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial. © 2012 Elsevier Ltd
On a conjecture of Hivert and Thiery about Steenrod operators
No full textWe prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics. In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics (F. Hivert and N. Thiéry, 2004 [HT]). In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture. © 2012 Elsevier Inc
Graph colorings, flows and arithmetic Tutte polynomial
No full textWe introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph.
We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte.We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte (1954) [9]. © 2012 Elsevier Inc
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