<i>G</i>-Tutte Polynomials and Abelian Lie Group Arrangements

Liu, Ye; Tran, Tan Nhat; Yoshinaga, Masahiko

doi:10.1093/imrn/rnz092

Cited by 22 publications

(41 citation statements)

References 44 publications

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“…. It is known that (e.g., [Ath96], [KTT08]) the 1-constituent f 1 A (t) coincides with χ A(R) (t) the characteristic polynomial (e.g., [OT92, Definition 2.52]) of the real hyperplane arrangement (or R-plexification in the sense of [LTY17]…”

Section: Preliminariesmentioning

confidence: 99%

“…Later on, Kamiya-Takemura-Terao showed that #M(A; Z ℓ , Z/qZ) is actually a quasi-polynomial in q [KTT08], and left the task of understanding the constituents of this quasi-polynomial to be an interesting problem. A number of attempts have been made in order to tackle the problem (e.g., [KTT08], [LTY17], [DFM17]), especially, two interpretations for every constituent via subspace and toric viewpoints have been found [TY18]. The mentioning establishments open a new direction for studying the combinatorics and topology of hyperplane and toric arrangements in one single quasi-polynomial.…”

Section: Introductionmentioning

confidence: 99%

“…The surprising connection between independent calculations on the Ehrhart quasi-polynomials [Sut98] and the characteristic quasi-polynomials [KTT07] produced a main flavor to the analysis on the deformations of root system arrangements in [Yos18b]. Combining the computation on the arithmetic Tutte polynomials of classical root systems [ACH15] with the previously mentioned calculations provided the authors in [LTY17] with the key observation of the identification between the last constituent of the characteristic quasi-polynomial and the corresponding toric arrangement.…”

Section: Introductionmentioning

confidence: 99%

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Characteristic quasi-polynomials of ideals and signed graphs of classical root systems

Tran

2019

European Journal of Combinatorics

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With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems (ABCD) with respect to the integer and root lattices. As a result, we obtain a full description of the characteristic polynomials of the toric arrangements defined by these ideals. As an application, we provide a combinatorial verification to the fact that the characteristic polynomial of every ideal subarrangement factors over the dual partition of the ideal in the classical cases.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characteristic quasi-polynomials of ideals and signed graphs of classical root systems

Tran

2019

European Journal of Combinatorics

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“…To capture them requires a richer combinatorial object, such as the matroids over Z of [27]. The requisite data also appear in the G-Tutte polynomial of [45], and Theorem 5.5 of that work explains how to recover from the G-Tutte polynomial every constituent of the characteristic quasi-polynomial, which is up to a sign the evaluation Q(1 − t, 0) of the Tutte quasi-polynomial. = m 1 (A) m 2 (A).…”

Section: Arithmetic Matroidsmentioning

confidence: 99%

Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

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This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.obtain several convolution formulae for different classes of combinatorial objects. Some of these formulae are well known, while others are (to the best of our knowledge) new: see for instance Propositions 5.12, 5.13 and 5.14 for the classical Tutte polynomial, Proposition 6.17 for the Las Vergnas polynomial, Proposition 6.20 for the Bollobás-Riordan polynomial and Theorem 10.9 for the arithmetic Tutte polynomial.We have chosen to exemplify our results with mostly matroid-like combinatorial structures, which are arguably simpler to deal with than topological examples. Our formalism, and in particular the mechanism of twist maps, should help uncover new invariants in the latter class and produce interesting convolution formulae. This will be the subject of a subsequent article.Layout. The structure of this paper is as follows. In Section 2 we give the definitions of minors systems and comonoids in set species. Section 3 contains the main results, including the definitions and statements of our universal invariants and formulae. Section 4 presents a number of further results which are less essential for our main developments.The remainder of the paper, Sections 5 through 10, comprises a sequence of applications of our theory to numerous individual minors systems. We work out Grothendieck groups, universal Tutte characters, and in many cases universal convolution formulae, and point out how these specialize to invariants and formulae present in the literature. Section 5 covers the minors system of matroids, of which all the subsequent sections are in one way or another generalizations, together with the minors system of graphs. Of the following sections, Section 6 is on delta-matroids, matroid perspectives, and their ilk is called on in Section 7 on relative matroids, but there are no (or at most incidental) dependences between these and Section 8 on polymatroids and generalised permutohedra, Section 9 on colored matroids, or Section 10 on arithmetic matroids, nor among the latter three, so the reader should have no trouble taking these in any order. The last four sections are also new by comparison with [37].Notation. We fix a commutative ring with unit K...

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“…This quasi-polynomial was defined to evaluate the cardinality of the complement of the q-reduced arrangement A(Z q ) in Z q . Then the characteristic polynomials of the hyperplane and toric arrangements coincide with the first and the last constituents of the characteristic quasi-polynomial, respectively ( [KTT08], [LTY17]).…”

Section: Introductionmentioning

confidence: 99%