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Representation Theory of Diagram Algebras: Subalgebras and Generalisations of the Partition Algebra

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Representation Theory of Diagram Algebras: Subalgebras and Generalisations of the Partition Algebra View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by White Rose E-theses Online Representation Theory of Diagram Algebras: Subalgebras and Generalisations of the Partition Algebra Chwas Abas Ahmed Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds School of Mathematics August 2016 ii iii The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. Work from the following jointly-authored preprint is included in this thesis: C. Ahmed, P. Martin and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids. arXiv preprint arXiv:1503.06718 Chapter 2 of this thesis is based on the above pre-print. The above work came after a visit of Prof. Mazorchuk (Uppsala University) to the Department of Mathematics in the University of Leeds in October 2014. The Mathematics in this paper comes from some discussions between the three authors after this visit. One third of the Mathematics of this work is contributed to me, and the online version of this preprint is agreed on by all the authors. However, Chapter 2 of this thesis is written entirely by me. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. c 2016 The University of Leeds and Chwas Abas Ahmed. iv v Acknowledgements I would like to show my deepest gratitude for my main supervisor Prof. Paul Martin for his patience and guidance through writing this thesis over the last four years. Without his constant support and valuable feedback it would not have been possible for me to produce this document. I would also like to thank my co-supervisor Dr. Alison Parker for her encouragement, support, and reading parts of this thesis. In addition, I would like to thank all the people in the School of Mathematics at the University of Leeds, especially the algebra group members and all the PhD students for making my time in Leeds more enjoyable, especially Raphael Bennett- Tennenhaus, Daniel Wolf, Ricardo Bello Aguirre and Daoud Siniora. I would like to thank Prof. Mazorchuk for the discussions that we had during writing the preprint version of Chapter 2 of this thesis. I gratefully acknowledge the comments on this thesis by both of my thesis examiners Prof. R. Marsh and Dr. R Gray. I also wish to thank the Kurdistan Regional Government and the University of Sulaimani for their financial support and for offering me an HCDP scholarship. Writing this document would not have been possible without my family's support. I wish to express my love and many thanks to my parents, my sister, my brothers and their families. vi Abstract This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions. vii To my family viii ix Contents Acknowledgements . .v Abstract . vi Dedication . vii Contents . ix List of figures . xi List of tables . xiii 1 Background 1 1.1 The partition algebra Pn(δ). ......................3 1.1.1 The basic partition category . .3 1.1.2 Partition diagrams . .4 1.1.3 The partition algebra Pn(δ) and some of its subalgebras. .6 1.1.4 Schur-Weyl duality for some subalgebras of the partition algebra. 11 1.2 Representation theory of Pn(δ) over the complex field. 13 1.3 Green's Relations. 18 1.4 Quasi-hereditary and Cellular algebras. 19 2 On the number of principal ideals in d-tonal partition monoids 25 d 2.1 Definition and order of the d-tonal monoid Pn.............. 27 d 2.2 Canonical representatives of the J -classes of Pn............ 30 d 2.3 Graded posets related to the set of all canonical elements of Pn.... 32 d 2.4 Relation between the set of canonical elements of Pn and I (ne1). 38 CONTENTS x 2.5 Relation between (I (ne1); ≤Xd ) and integer partition combinatorics. 41 2.6 Combinatorial results about Pd, for small values of d.......... 46 2.6.1 The case d =1........................... 46 2.6.2 The case d =2........................... 47 2.6.3 The case d =3........................... 48 2.7 Relation with Hollow Hexagons by using reflection groups. 55 2.7.1 T-hexagons and their H-envelope. 58 2.7.2 Characters of t-hexagons . 64 2.7.3 Elementary operations on distinguished t-hexagons. 66 d 3 Representation theory of the planar d-tonal partition algebra Tn (δ). 71 d 3.1 Definition of the planar d-tonal partition monoid Tn and the planar d d-tonal algebra Tn (δ)............................ 73 d 3.2 On the order of Tn ............................. 73 d 3.3 The J -classes of Tn ............................ 75 2 2 3.4 Relation between Tn (δ ) and the Fuss-Catalan algebra. 81 d 3.5 An indexing set for the simple modules of Tn (δ). 85 d 3.6 On the quasi-heredity and Cellularity of Tn (δ). 90 d 3.7 On the standard modules of Tn (δ). ................... 94 d 3.7.1 Construction of the standard modules of Tn (δ)......... 94 d 3.7.2 Restriction rules for the standard modules of Tn (δ). 98 d 3.8 On the semisimple cases of Tn (δ)..................... 102 d 3.8.1 Generic Semisimplicity of Tn (δ). 102 d 3.8.2 The Bratteli diagram for fTn (δ)gn in the semisimple cases. 106 d ⊗n 3.8.3 Action of Tn (δ) on V ...................... 108 2 3.9 A necessary and sufficient condition for the semisimplicity of Tn (δ). 110 Bibliography 129 xi List of figures 1.1 Example of the product of partition diagrams . .5 1.2 Horizontal product of diagrams . .5 1.3 The elements uk and mk .........................6 1.4 A poset describes the embedding of some subalgebras of the partition algebra. 10 2 2.1 An example and a non-example of an elements in P6 ......... 27 3 2.2 The diagram a is a canonical element in P13 while the digram b is not 2 a canonical element in P12........................ 30 2.3 Case d = 3, I (14e1) poset . 36 2.4 Quotient of the poset Π7 by the equivalence relation ∼2........ 44 (h) 2.5 Triangular arrangement of Ωd ..................... 52 2.6 The alcove system of Ae2......................... 57 2.7 The dual graph of the the alcove system of Ae2............. 58 2.8 An example of a coronoid system. 59 2.9 Triangles marked with ? form a tiling strip of type 1. 60 2.10 The bold convex polygon is a t-hexagon and its hexagonal envelope is the doted line shape. 61 2.11 Basic vectors and lines . 64 LIST OF FIGURES xii 3.1 An example describing the bijection in the Proposition 3.2.2, with n =6. ................................... 74 3.2 The first equality illustrates moving a northern non-propagating part to left, and the second equality explains gluing a non-propagating part to an other part. 78 3.3 The covering relation l0, when d = 2 and n = 5. 80 2 3.4 An example of two different J -equivalent elements in T8 , describing Lemma 3.3.8. 81 3.5 An element of F(6; 1)........................... 82 3.6 An element of F(6; 1)........................... 83 3.7 The idempotent ei;d, where 0 ≤ i < bn=dc ................ 86 3.8 Right restriction rule, for n ≤ 4 and d =2 ............... 101 d 3.9 The element wi 2 Tn , where 0 ≤ i < n − d ............... 114 xiii List of tables d 2.1 Cardinality of Pn, for some small values of d and n.......... 29 (d) 2.2 Some values of In ; n ≤ 16 and d ≤ 4.................. 42 d 3.1 Cardinality of Tn , for some small values of d and n.......... 76 1 Chapter 1 Background We devote this chapter to set up the notation of the thesis and recall some necessary background for the later chapters. While we are developing the notation, we state the aims of this thesis, and record some of the history of the development of this area of research. In section 1 we define the partition algebras. Roughly speaking the partition algebras are a tower of finite dimensional algebras that possess a basis which can be described by diagrams.
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