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Prof Robert J Marsh

Professor of Pure Mathematics
Pure Mathematics

Contact details

Room: 8.08
Tel: +44 (0)113 3435164
Email: R.J.Marsh @ leeds.ac.uk

Keywords

Cluster algebras
Representations of algebras
Triangulated categories
Homological algebra
Tilting theory
Algebraic groups
Algebraic combinatorics
Coxeter groups

Research interests

My research focuses on the representation theory of finite dimensional algebras, homological algebra and triangulated categories. A key aspect of this is the new representation theory arising from cluster algebras, which were introduced in order to model the canonical basis of a quantum group. In particular I am interested in the use of the combinatorial geometry of Riemann surfaces to model categories arising in representation theory. I am also interested in the representation theory of diagram algebras, which arise in the study of transfer matrices in statistical mechanics.

There are strong combinatorial aspects to the theory and some of my work is involved in investigating this, including links between the two areas and applications to discrete integrable systems.

Useful links

Personal home page
Cluster algebras portal
What is ... a Cluster Algebra? An article by Andrei Zelevinsky.
I am visiting Kings College London until July 2014.

Publications

Marsh RJ, Palu Y Nearly Morita equivalences and rigid objects Nagoya Mathematical Journal, 225, 64-99, 2017
DOI:10.1017/nmj.2016.27
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Grant J, Marsh RJ Braid groups and quiver mutation Pacific Journal of Mathematics, 290, 77-116, 2017
DOI:10.2140/pjm.2017.290.77
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Marsh RJ, Reiten I Rigid and Schurian modules over cluster-tilted algebras of tame type Mathematische Zeitschrift, 284, 643-682, 2016
DOI:10.1007/s00209-016-1668-z
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Baur K, King A, Marsh RJ Dimer models and cluster categories of Grassmannians Proceedings of the London Mathematical Society, 113, 213-260, 2016
DOI:10.1112/plms/pdw029
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Marsh RJ, Scott JS Twists of Plücker coordinates as dimer partition functions Communications in Mathematical Physics, 341, 821-884, 2016
DOI:10.1007/s00220-015-2493-7
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Barot M, Marsh RJ Reflection group presentations arising from cluster algebras. Transactions of the American Mathematical Society, 367, 1945-1967, 2015
DOI:10.1090/S0002-9947-2014-06147-3
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Marsh RJ, Schroll S The geometry of Brauer graph algebras and cluster mutations. Journal of Algebra, 419, 141-166, 2014
DOI:10.1016/j.jalgebra.2014.08.002
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Baur K, Buan AB, Marsh RJ Torsion pairs and rigid objects in tubes Algebras and Representation Theory, 17, 565-591, 2014
DOI:10.1007/s10468-013-9410-6
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Marsh RJ, Schroll S A circular order on edge-coloured trees and RNA m-diagrams Advances in Applied Mathematics, 54, 11-26, 2014
DOI:10.1016/j.aam.2013.09.009
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Marsh RJ, Palu Y Coloured quivers for rigid objects and partial triangulations: The unpunctured case. Proceedings of the London Mathematical Society, 108, 411-440, 2014
DOI:10.1112/plms/pdt032
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Buan AB, Marsh RJ From triangulated categories to module categories via localisation Transactions of the American Mathematical Society, 365, 2845-2861, 2013
DOI:10.1090/S0002-9947-2012-05631-5
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Marsh RJ Lecture Notes on Cluster Algebras. 2013

Bakke Buan A, Marsh RJ Erratum: From triangulated categories to module categories via localisation II: Calculus of fractions (Journal of the London Mathematical Society (2012) 86 (152-170)) Journal of the London Mathematical Society, 87, 643, 2013
DOI:10.1112/jlms/jdt015

Baur K, Marsh RJ A geometric model of tube categories Journal of Algebra, 362, 178-191, 2012
DOI:10.1016/j.jalgebra.2012.04.009
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Baur K, Marsh RJ Categorification of a frieze pattern determinant Journal of Combinatorial Theory. Series A, 119, 1110-1122, 2012
DOI:10.1016/j.jcta.2012.02.006
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Marsh RJ, Buan AB From triangulated categories to module categories via localisation II: calculus of fractions. Journal of the London Mathematical Society, 86, 152-170, 2012
DOI:10.1112/jlms/jdr077
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Fordy AP, Marsh RJ Cluster mutation-periodic quivers and associated Laurent sequences Journal of Algebraic Combinatorics, 34, 19-66, 2011
DOI:10.1007/s10801-010-0262-4

Marsh RJ, Martin P Tiling bijections between paths and Brauer diagrams Journal of Algebraic Combinatorics, 33, 427-453, 2011
DOI:10.1007/s10801-010-0252-6

Buan AB, Marsh RJ, Vatne DF Cluster structures from 2-Calabi-Yau categories with loops Mathematische Zeitschrift, 265, 951-970, 2010
DOI:10.1007/s00209-009-0549-0

Buan AB, Marsh RJ Denominators in cluster algebras of affine type Journal of Algebra, 323, 2083-2102, 2010
DOI:10.1016/j.jalgebra.2010.02.012

Baur K, Marsh RJ Frieze patterns for punctured discs Journal of Algebraic Combinatorics, 30, 349-379, 2009
DOI:10.1007/s10801-008-0161-0

Buan AB, Marsh RJ, Reiten I Denominators of cluster variables Journal of the London Mathematical Society, 79, 589-611, 2009
DOI:10.1112/jlms/jdn082

Buan AB, Marsh RJ, Reiten I Cluster mutation via quiver representations Commentarii Mathematici Helvetici, 83, 143-177, 2008

Baur K, Marsh RJ A geometric description of m-cluster categories Transactions of the American Mathematical Society, 360, 5789-5803, 2008

Proceedings of the Rings and Modules Conference, Lisbon, Portugal, 16-19.09.2008, in Honour of Patrick F. Smith's 65th Birthday. 2008

Buan AB, Marsh RJ, Reiten I, Todorov G Clusters and seeds in acyclic cluster algebras Proceedings of the American Mathematical Society, 135, 3049-3060, 2007
DOI:10.1090/S0002-9939-07-08801-6
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Baur K, Marsh RJ A geometric description of the m-cluster categories of type Dn International Mathematics Research Notices, 2007
DOI:10.1093/imrn/rnm011
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Buan AB, Marsh RJ, Reiten I Cluster-tilted algebras Transactions of the American Mathematical Society, 359, 323-332, 2007
DOI:10.1090/S0002-9947-06-03879-7
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Buan AB, Marsh RJ, Reiten I, Todorov G Clusters and seeds in acyclic cluster algebras Proceedings of the American Mathematical Society, 135, 323-332, 2007
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Buan AB, Marsh RJ, Reineke M, Reiten I, Todorov G Tilting theory and cluster combinatorics Advances in Mathematics, 204, 572-618, 2006
DOI:10.1016/j.aim.2005.06.003
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Buan AB, Marsh RJ, Reiten I Cluster-tilted algebras of finite representation type. Journal of Algebra, 306, 412-431, 2006
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Buan AB; Marsh RJ Cluster-tilting theoryTrends in Representation Theory of Algebras and Related Topics, Contemporary Mathematics, 1-30 2006
DOI:10.1090/conm/406
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Marsh RJ, Rietsch K Parametrizations of flag varieties Representation Theory, 8, 212-242, 2004
DOI:10.1090/S1088-4165-04-00230-4
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Caldero P, Marsh RJ A multiplicative property of quantum flag minors II Journal of London Mathematical Society, 69, 608-622, 2004
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Caldero P, Marsh RJ, Morier-Genoud S Realisation of Lusztig cones Representation Theory, 8, 458-478, 2004
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Marsh RJ, Reineke M, Zelevinsky A Generalised associahedra via quiver representations Transactions of the American Mathematical Society, 355, 4171-4186, 2003
DOI:10.1090/S0002-9947-03-03320-8
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Marsh RJ, Reineke M Canonical basis linearity regions arising from quiver representations Journal of Algebra, 270, 696-727, 2003
DOI:10.1016/j.jalgebra.2003.04.002
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Marsh RJ Rectangle diagrams for the Lusztig cones of quantized enveloping algebras of type A Journal of Algebra, 249, 377-401, 2002
DOI:10.1006/jabr.2001.9031
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Marsh RJ The Lusztig cones of quantized enveloping algebras of type A Journal of Algebra, 244, 59-75, 2002
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Marsh RJ, Rietsch K The intersection of opposed big cells in the real flag variety of type G_2 Proceedings of the London Mathematical Society, 85, 22-42, 2002
DOI:10.1112/S0024611502013539
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Carter RW, Marsh RJ Canonical bases and piecewise-linear combinatorics In Quantum groups and Lie theory, 290, 33-47, 2001
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Carter RW, Marsh RJ Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A4 Journal of Algebra, 234, 545-603, 2000
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Marsh RJ Canonical bases and standard monomials Proceedings of the Edinburgh Mathematical Society, 41, 611-623, 1998
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Marsh RJ More tight monomials for quantized enveloping algebras. Journal of Algebra, 204, 711-732, 1998

Green RM, Marsh RJ Quantized symmetric powers Quarterly Journal of Mathematics, 48, 479-492, 1997

Marsh RJ Canonical bases for fundamental modules of quantized enveloping algebras of type A Communications in Algebra, 25, 1365-1386, 1997

Marsh RJ Canonical bases for the miniscule modules of the quantized enveloping algebras of type B and D Communications in Algebra, 25, 1387-1411, 1997

Marsh RJ Algorithms to obtain the canonical basis in some fundamental modules of quantum groups Journal of Algebra, 186, 831-860, 1996