Home > Research > Publications & Outputs > Applications of graded methods to cluster varia...

Electronic data

  • 2017Booker-PricePhD

    Final published version, 0.99 MB, PDF document

    Available under license: CC BY-ND: Creative Commons Attribution-NoDerivatives 4.0 International License

  • 2017Booker-Price_code

    Final published version, 94.5 KB, application/octet-stream

    Available under license: CC BY: Creative Commons Attribution 4.0 International License

  • 2017Booker-Price_Code_Documentation

    Final published version, 141 KB, PDF document

Text available via DOI:

View graph of relations

Applications of graded methods to cluster variables in arbitrary types

Research output: ThesisDoctoral Thesis

Published

Standard

Applications of graded methods to cluster variables in arbitrary types. / Booker-Price, Thomas.
Lancaster University, 2017. 136 p.

Research output: ThesisDoctoral Thesis

Harvard

APA

Vancouver

Booker-Price T. Applications of graded methods to cluster variables in arbitrary types. Lancaster University, 2017. 136 p. doi: 10.17635/lancaster/thesis/151

Author

Bibtex

@phdthesis{ce5c9954a127475ab089433a5e72de68,
title = "Applications of graded methods to cluster variables in arbitrary types",
abstract = "This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We start by considering two classes of finite type cluster algebras: those of type Bn and Cn. We give the number of cluster variables of each occurring degree and verify that the grading is balanced. These results complete a classification in [16] for coefficient-free finite type cluster algebras.We then consider gradings on cluster algebras generated by 3×3 skew-symmetric matrices. We show that the mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables (excepting one particular case). For the mutation-acyclic matrices, we prove that all occurring degrees have infinitely many variables and give a direct proof that the gradings are balanced.We provide a condition for a graded cluster algebra generated by a quiver to have infinitely many degrees, based on the presence of a subquiver in its mutation class. We use this to study the gradings on cluster algebras that are (quantum) coordinate rings of matrices and Grassmannians and show that they contain cluster variables of all degrees in N.Next we consider the finite list (given in [9]) of mutation-finite quivers that do not correspond to triangulations of marked surfaces. We show that A(X7) has a grading in which there are only two degrees, with infinitely many cluster variables in both. We also show that the gradings arising from Ee6, Ee7 and Ee8 have infinitely many variables in certain degrees.Finally, we study gradings arising from triangulations of marked bordered 2- dimensional surfaces (see [10]). We adapt a definition from [24] to define the space of valuation functions on such a surface and prove combinatorially that this space is isomorphic to the space of gradings on the associated cluster algebra. We illustrate this theory by applying it to a family of examples, namely, the annulus with n + m marked points. We show that the standard grading is of mixed type, with finitely many variables in some degrees and infinitely many in the others. We also give an alternative grading in which all degrees have infinitely many cluster variables.",
keywords = "cluster algebra, graded cluster algebra, infinite type, grading",
author = "Thomas Booker-Price",
year = "2017",
doi = "10.17635/lancaster/thesis/151",
language = "English",
publisher = "Lancaster University",
school = "Lancaster University",

}

RIS

TY - BOOK

T1 - Applications of graded methods to cluster variables in arbitrary types

AU - Booker-Price, Thomas

PY - 2017

Y1 - 2017

N2 - This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We start by considering two classes of finite type cluster algebras: those of type Bn and Cn. We give the number of cluster variables of each occurring degree and verify that the grading is balanced. These results complete a classification in [16] for coefficient-free finite type cluster algebras.We then consider gradings on cluster algebras generated by 3×3 skew-symmetric matrices. We show that the mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables (excepting one particular case). For the mutation-acyclic matrices, we prove that all occurring degrees have infinitely many variables and give a direct proof that the gradings are balanced.We provide a condition for a graded cluster algebra generated by a quiver to have infinitely many degrees, based on the presence of a subquiver in its mutation class. We use this to study the gradings on cluster algebras that are (quantum) coordinate rings of matrices and Grassmannians and show that they contain cluster variables of all degrees in N.Next we consider the finite list (given in [9]) of mutation-finite quivers that do not correspond to triangulations of marked surfaces. We show that A(X7) has a grading in which there are only two degrees, with infinitely many cluster variables in both. We also show that the gradings arising from Ee6, Ee7 and Ee8 have infinitely many variables in certain degrees.Finally, we study gradings arising from triangulations of marked bordered 2- dimensional surfaces (see [10]). We adapt a definition from [24] to define the space of valuation functions on such a surface and prove combinatorially that this space is isomorphic to the space of gradings on the associated cluster algebra. We illustrate this theory by applying it to a family of examples, namely, the annulus with n + m marked points. We show that the standard grading is of mixed type, with finitely many variables in some degrees and infinitely many in the others. We also give an alternative grading in which all degrees have infinitely many cluster variables.

AB - This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We start by considering two classes of finite type cluster algebras: those of type Bn and Cn. We give the number of cluster variables of each occurring degree and verify that the grading is balanced. These results complete a classification in [16] for coefficient-free finite type cluster algebras.We then consider gradings on cluster algebras generated by 3×3 skew-symmetric matrices. We show that the mutation-cyclic matrices give rise to gradings in which all occurring degrees are positive and have only finitely many associated cluster variables (excepting one particular case). For the mutation-acyclic matrices, we prove that all occurring degrees have infinitely many variables and give a direct proof that the gradings are balanced.We provide a condition for a graded cluster algebra generated by a quiver to have infinitely many degrees, based on the presence of a subquiver in its mutation class. We use this to study the gradings on cluster algebras that are (quantum) coordinate rings of matrices and Grassmannians and show that they contain cluster variables of all degrees in N.Next we consider the finite list (given in [9]) of mutation-finite quivers that do not correspond to triangulations of marked surfaces. We show that A(X7) has a grading in which there are only two degrees, with infinitely many cluster variables in both. We also show that the gradings arising from Ee6, Ee7 and Ee8 have infinitely many variables in certain degrees.Finally, we study gradings arising from triangulations of marked bordered 2- dimensional surfaces (see [10]). We adapt a definition from [24] to define the space of valuation functions on such a surface and prove combinatorially that this space is isomorphic to the space of gradings on the associated cluster algebra. We illustrate this theory by applying it to a family of examples, namely, the annulus with n + m marked points. We show that the standard grading is of mixed type, with finitely many variables in some degrees and infinitely many in the others. We also give an alternative grading in which all degrees have infinitely many cluster variables.

KW - cluster algebra

KW - graded cluster algebra

KW - infinite type

KW - grading

U2 - 10.17635/lancaster/thesis/151

DO - 10.17635/lancaster/thesis/151

M3 - Doctoral Thesis

PB - Lancaster University

ER -