The algebras A(Γ), where Γ is a directed layered graph, were first constructed by I. Gelfand, S. ... more The algebras A(Γ), where Γ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson. These algebras are generalizations of the algebras Qn, which are related to factorizations of non-commutative polynomials. It was conjectured that these algebras were Koszul. In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul A(Γ) corresponding to a graph Γ with 18 edges and 11 vertices. We produce an example of a directed layered graph Γ with 13 edges and 9 vertices which produces a non-Koszul A(Γ). We also show this is the minimal example with this property.
Noncommutative Birational Geometry, Representations and Combinatorics, 2013
Copying and reprinting. Material in this book may be reproduced by any means for educational and ... more Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society,
The algebras Q_n describe the relationship between the roots and coefficients of a non-commutativ... more The algebras Q_n describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the graph K_3. We also show this algebra is Koszul using the lattice definition.
Summary In his popular blog, Brian Hayes presented a four-by-four KenKen puzzle over a set of com... more Summary In his popular blog, Brian Hayes presented a four-by-four KenKen puzzle over a set of complex numbers, mentioning that he had not checked the uniqueness of the solution. Here, we study all cage patterns over similar puzzles that allow for unique solutions and show that they are equivalent to Hayes's choice. We define a group action that preserves the size of solution sets and proceed to classify all puzzles with unique solutions before concluding with suggested activities and several open questions.
We produce an example of a directed layered graph $\Gamma$ with 13 edges and 9 vertices which pro... more We produce an example of a directed layered graph $\Gamma$ with 13 edges and 9 vertices which produces a non-Koszul $A(\Gamma)$. We also show this is the minimal example with this property.
The algebras $Q_n$ describe the relationship between the roots and coefficients of a non-commutat... more The algebras $Q_n$ describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the graph $K_3$. We also show this algebra is Koszul using the lattice definition.
The algebras $Q_n$ describe the relationship between the roots and coefficients of a non-commutat... more The algebras $Q_n$ describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the $n$-vertex path, $P_n$. We also show this algebra is Koszul. We do this by first looking at class of quadratic algebras we call Partially Generator Commuting. We then find a sufficient condition for a PGC-Algebra to be Koszul and use this to show a similar class of PGC algebras, which we call ch$P_n$, is Koszul. Then we show it is possible to extend what we did to the algebras $P_n$ although they are not PGC. Finally we examine the Hilbert Series of the algebras $P_n$
The algebras Q n describe the relationship between the roots and coefficients of a non-commutativ... more The algebras Q n describe the relationship between the roots and coefficients of a non-commutative polynomial. I. Gelfand, S. Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this paper we find the Hilbert series of the class of algebras corresponding to the graph K 3 and show that this algebra is Koszul.
The algebras Qn describe the relationship between the roots and coefficients of a non-commutative... more The algebras Qn describe the relationship between the roots and coefficients of a non-commutative polynomial. I.Gelfand, S.Gelfand, and V. Retakh have defined quotients of these algebras corresponding to graphs. In this work we find the Hilbert series of the class of algebras corresponding to the graph K3. We also show this algebra is Koszul.
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