Table Of ContentVrije Universiteit Brussel
Faculty of Science and Bio-engineering Sciences
Identities of Affine Algebras
and their asymptotic behaviour
Geoffrey Janssens
Mathematics Department
Promotor: Prof. Dr. Eric Jespers
Co-Promotor: Dr. Alexey Gordienko
Doctoral thesis submitted in partial fulfillment of the requirements for
the degree of Doctor in Science
March 2017
Pure mathematics is, in its way,
the poetry of logical ideas.
Albert Einstein
This thesis was defended at VUB, in Brussels, behind closed doors on May 9th and
publicly on May 19th 2017. The members of the jury were:
President: Prof. Stefaan Caenepeel,Vrije Universiteit Brussel, Belgium
Secretary: Prof. Kenny De Commer,Vrije Universiteit Brussel, Belgium
Advisor: Prof. Eric Jespers,Vrije Universiteit Brussel, Belgium
Co-Advisor: Dr. Alexey Gordienko,Vrije Universiteit Brussel, Belgium
Dr. Andreas Bächle,Vrije Universiteit Brussel, Belgium
Dr. Špela Špenko,Vrije Universiteit Brussel, Belgium
Prof. Ann Dooms,Vrije Universiteit Brussel, Belgium
Prof. Dominique Maes,Vrije Universiteit Brussel, Belgium
Prof. Eli Aljadeff,Technion-Israel Institute of Technology, Israel
Prof. Mikhail Zaicev,Moscow State University, Russia
Vrije Universiteit Brussel
Faculteit Wetenschappen en Bio-ingenieurswetenschappen
Identities of Affine Algebras
and their asymptotic behaviour
Geoffrey Janssens
Vakgroep Wiskunde
Promotor: Prof. Eric Jespers
Co-Promotor: Dr. Alexey Gordienko
Proefschrift voorgelegd aan de Faculteit Wetenschappen,
voor het verkrijgen van de graad van Doctor in de Wetenschappen
Mei 2017
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Contents
Acknowledgements v
Notational Conventions viii
Introduction 1
1 A glimpse into the theory of Polynomial Identities 13
1.1 From Polynomials to Asymptotics . . . . . . . . . . . . . . . . . . . . . . 14
1.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.2 Codimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Representation theory of S . . . . . . . . . . . . . . . . . . . . . . . . . . 24
n
1.2.1 Specht modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.2 p-irreducibility of Specht modules. . . . . . . . . . . . . . . . . . . 30
1.2.3 Dimension of Specht modules . . . . . . . . . . . . . . . . . . . . . 31
1.2.4 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.2.5 Specht Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3 Behind the Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.3.1 Amitsur and Regev Conjecture . . . . . . . . . . . . . . . . . . . . 41
1.3.2 On computing the Exponential Growth Rate . . . . . . . . . . . . 45
1.3.3 Some further remarks . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.4 Kemer Theory and Representability . . . . . . . . . . . . . . . . . . . . . 52
1.4.1 Basic algebras and Kemer polynomials . . . . . . . . . . . . . . . . 52
i
1.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2 The Polynomial Growth Rate of Codimensions 60
2.1 Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.1.1 Some reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.1.2 The key lemma and upper bound . . . . . . . . . . . . . . . . . . . 68
2.2 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3 Codimension theory for rings 82
3.1 Codimensions of Torsion-free rings . . . . . . . . . . . . . . . . . . . . . . 84
3.2 A chain from Ordinary to Proper Polynomial Functions . . . . . . . . . . 87
3.3 A particular case of the Littlewood-Richardson rule . . . . . . . . . . . . . 89
3.4 Algebras of Upper Triangular Matrices . . . . . . . . . . . . . . . . . . . . 90
3.4.1 Codimensions and multilinear identities . . . . . . . . . . . . . . . 91
3.4.2 ZS -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
n
3.5 Grassmann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5.1 Codimensions and polynomial identities . . . . . . . . . . . . . . . 95
3.5.2 ZS -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
n
4 Semigroup Graded-Simple Algebras 98
4.1 General results and Reducing the Grading . . . . . . . . . . . . . . . . . . 100
4.2 Left ideals of matrix algebras . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Graded-simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Existence theorems for graded-simple algebras . . . . . . . . . . . . . . . . 117
5 Exponential Growth of Semigroup Graded Algebras 122
5.1 Introduction to Graded and Generalized Codimensions . . . . . . . . . . . 123
5.2 Overview of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Upper bound for T-graded-simple algebras . . . . . . . . . . . . . . . . . . 130
5.4 The case A/J(A) ∼= M (F) and PIexpT-gr(A) = dimA . . . . . . . . . . . 139
2
5.5 The case A/J(A) ∼= M (F) and PIexpT-gr(A) < dimA . . . . . . . . . . . 145
2
5.6 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.6.1 Survey on Graded Lie Polynomials and their Codimensions . . . . 157
5.7 A graded non-integer Exponent . . . . . . . . . . . . . . . . . . . . . . . . 161
5.7.1 Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
ii
5.7.2 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.8 Semisimple Lie algebras with generalized action . . . . . . . . . . . . . . . 170
5.8.1 H-simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Inleiding (Nederlands) 176
Bibliography 189
Index 199
iii
Description:that are isomorphic to S(λ)/mS(λ), where λ is a partition of n and m ∈ Z is related to the torsion of R? by certain partitions, called p-regular, and to each corresponds the irreducible represen- tation DF (λ) = SF (λ)/Rad(SF (λ)), where Dit is een zeer rijke theorie. De essentie wordt he
