| 1. |
- Crispin Quiñonez, Veronica, et al.
(författare)
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On ideals generated by two generic quadratic forms in the exterior algebra
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.
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| 2. |
- Nenashev, Gleb, 1992-
(författare)
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Around power ideals : From Fröberg's conjecture to zonotopal algebra
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we study power algebras, which are quotient of polynomial rings by power ideals. We will study Hilbert series of such ideals and their other properties. We consider two important special cases, namely, zonotopal ideals and generic ideals. Such ideals have a lot combinatorial properties.In the first chapter we study zonotopal ideals, which were defined and used in several earlier publications. The most important works are by F.Ardila and A.Postnikov and by O.Holtz and A.Ron. These papers originate from different sources, the first source is homology theory, the second one is the theory of box splines. We study quotient algebras by these ideals; these algebras have a nice interpretation for their Hilbert series, as specializations of their Tutte polynomials. There are two important subclasses of these algebras, called unimodular and graphical. The graphical algebras were defined by A.Postnikov and B.Shapiro. In particular, the external algebra of a complete graph is exactly the algebra generated by the Bott-Chern forms of the corresponding complete flag variety. One of the main results of the thesis is a characterization of external algebras. In fact, for the case of graphical and unimodular algebras we prove that external algebras are in one-to-one correspondence with graphical and regular matroids, respectively.In the second chapter we study Hilbert series of generic ideals. By a generic ideal we mean an ideal generated by forms from some class, whose coefficients belong to a Zariski-open set. There are two main classes to consider: the first class is when we fix the degrees of generators; the famous Fröberg's conjecture gives the expected Hilbert series of such ideals; the second class is when an ideal is generated by powers of generic linear forms. There are a few partial results on Fröberg's conjecture, namely, when the number of variables is at most three. In both classes the Hilbert series is known in the case when the number of generators is at most (n+1). In both cases we construct a lot of examples when the degree of generators are the same and the Hilbert series is the expected one.
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| 3. |
- Nenashev, Gleb, 1992-
(författare)
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On a class of commutative algebras associated to graphs
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In 2004 Alexander Postnikov and Boris Shapiro introduced a class of commutative algebras for non-directed graphs. There are two main types of such algebras, algebras of the first type count spanning trees and algebras of the second type count spanning forests. These algebras have a number of interesting properties including an explicit formula for their Hilbert series. In this thesis we mainly work with the second type of algebras, we discover more properties of the original algebra and construct a few generalizations. In particular we prove that the algebra counting forests depends only on graphical matroid of the graph and converse. Furthermore, its "K-theoretic" filtration reconstructs the whole graph. We introduse $t$ labelled algebras of a graph, their Hilbert series contains complete information about the Tutte polynomial of the initial graph. Finally we introduce similar algebras for hypergraphs. To do this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid.
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| 4. |
- Nenashev, Gleb, 1992-, et al.
(författare)
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Unimodular Zonotopal Algebra
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- For any given hyperplane arrangment, we introduce Q-deformations of its external zonotopal algebras. Furthermore, we introduce Hecke deformations of all three types of unimodular zonotopal algebras, which gives a new definition of non-deformed central and internal zonotopal algebras.
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| 5. |
- Kirillov, Anatol N., et al.
(författare)
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On Q-Deformations of Postnikov-Shapiro Algebras
- 2017
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Ingår i: Proceedings of the 29th Conference on Formal Power Series and Algebraic Combinatorics. - Strasbourg : Séminaire Lotharingien de Combinatoire.
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Konferensbidrag (refereegranskat)abstract
- For any given loopless graph G, we introduce Q - deformations of its Postnikov-Shapiro algebras counting spanning trees and forests. We determine the total dimension of the algebras; our proof also gives a new proof of the formula for the total dimensions of the usual Postnikov-Shapiro algebras.
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| 6. |
- Nenashev, Gleb, 1992-
(författare)
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Classification of external Zonotopal algebras
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids. For the case of central algebras we formulate a conjecture.
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| 7. |
- Nenashev, Gleb, et al.
(författare)
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"K-theoretic" analog of Postnikov-Shapiro algebra distinguishes graphs
- 2017
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Ingår i: Journal of combinatorial theory. Series A (Print). - : Elsevier BV. - 0097-3165 .- 1096-0899. ; 148, s. 316-332
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study a filtered "K-theoretical" analog of a graded algebra associated to any loopless graph G which was introduced in \cite{PS}. We show that two such filtered algebras are isomorphic if and only if their graphs are isomorphic. We also study a large family of filtered generalizations of the latter graded algebra which includes the above "K-theoretical" analog.
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| 8. |
- Nenashev, Gleb
(författare)
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On Postnikov-Shapiro Algebras and their generalizations
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- A.Postnikov and B.Shapiro introduced a class of commutative algebras which enumerate forests and trees of graphs. Our main result is that the algebra counting forests depends only on graphical matroid and converse.Also we generalize algebras for a hypergraph. For this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid.
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| 9. |
- Nenashev, Gleb
(författare)
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Postnikov-Shapiro Algebras, Graphical Matroids and their generalizations
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we consider the original and different generalizations of Postnikov-Shapiro algebra which enumerate forests and trees of graphs, see~\cite{PSh}. Our main result is that the algebra counting forests depends only on graphical matroid and converse. Also we generalize algebras for a hypergraph. For this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid. We present 3 different equivalent definitions of spanning forests and trees, which can be read independently from other parts of the paper.
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