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Algebras associated to acyclic directed graphs. (English) Zbl 1177.05127

Summary: We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices and edges. Each finite acyclic directed graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.

MSC:

05E05 Symmetric functions and generalizations
05C20 Directed graphs (digraphs), tournaments
15A15 Determinants, permanents, traces, other special matrix functions
16T30 Connections of Hopf algebras with combinatorics
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[1] Anick, D. J., Non-commutative graded algebras and their Hilbert series, J. Algebra, 78, 120-140 (1982) · Zbl 0502.16002
[2] Duffy, C., Representations of \(Aut(A(\Gamma))\) acting on homogeneous components of \(A(\Gamma)\) and \(A(\Gamma)^! (2008)\), Adv. in Appl. Math., in press · Zbl 1173.16014
[3] C. Duffy, Graded traces and irreducible representations of \(\mathit{Aut}(A(\operatorname{\Gamma;}))A(\operatorname{\Gamma;})A ( \operatorname{\Gamma;} )^!\); C. Duffy, Graded traces and irreducible representations of \(\mathit{Aut}(A(\operatorname{\Gamma;}))A(\operatorname{\Gamma;})A ( \operatorname{\Gamma;} )^!\)
[4] Gelfand, I.; Gelfand, S.; Retakh, V.; Wilson, R., Factorizations of polynomials over noncommutative algebras and sufficient sets of edges in directed graphs, Lett. Math. Phys., 74, 153-167 (2005) · Zbl 1112.05048
[5] Gelfand, I.; Retakh, V.; Serconek, S.; Wilson, R., On a class of algebras associated to directed graphs, Selecta Math. (N.S.), 11, 281-295 (2005) · Zbl 1080.05040
[6] Gelfand, I.; Retakh, V.; Wilson, R., Quadratic-linear algebras associated with decompositions of noncommutative polynomials and noncommutative differential polynomials, Selecta Math., 7, 493-523 (2001) · Zbl 0992.16025
[7] Golod, E. S., Noncommutative complete intersections and the homology of the Shafarevich complex, Russian Math. Surveys, 52, 4, 830-831 (1997) · Zbl 0926.16009
[8] Retakh, V.; Serconek, S.; Wilson, R., On a class of Koszul algebras associated to directed graphs, J. Algebra, 304, 1114-1129 (2006) · Zbl 1127.16022
[9] Retakh, V.; Serconek, S.; Wilson, R., Hilbert series of algebras associated to directed graphs, J. Algebra, 312, 142-151 (2007) · Zbl 1139.16015
[10] Retakh, V.; Serconek, S.; Wilson, R., Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials, (Proceedings of the Conference “Lie Algebras, Vertex Operator Algebras and Their Applications”. Proceedings of the Conference “Lie Algebras, Vertex Operator Algebras and Their Applications”, Contemp. Math., vol. 412 (2007)), 201-219 · Zbl 1138.05324
[11] Stanley, R. P., Enumerative Combinatorics, vol. 1 (1999), Cambridge University Press · Zbl 1247.05003
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