# School of Mathematics

## 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 theory*Trends 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

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