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Deligne categories and reduced Kronecker coefficients
Get PDFThe Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ,μ,τ of n, the multiplicity of λ in μ⊗τ is called the Kronecker coefficient g[superscript λ][subscript μ,τ]. When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood–Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories [bar under Rep](S[subscript t]). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories [bar under Rep](S[subscript t]) and derive new combinatorial identities
Schur Weyl duality in complex rank
No full textThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 207-208).This thesis gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1) we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(St) equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(St) and a localization of the category Op/t,v (the parabolic category 0 for gl(V) associated with the pair (V, 1)). In particular, we obtain an exact contravariant functor SWt from the category Repab(St) (the abelian envelope of the category Rep(St)) to a certain quotient of the category Op/t v. This quotient, denoted by 0 p/t v, is obtained by taking the full subcategory of Op/t v consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the contravariant functor SWt makes Op/t v a Serre quotient of the category Repab(St)OP, and the kernel of SWt can be explicitly described. In the second part of this thesis, we consider the case when V = C[infinity] . We define the appropriate version of the parabolic category 0 and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories Op/t1CN with N tending to infinity. The Schur-Weyl functors SWt,CN then give an anti-equivalence between the category Op[infinity]/t C[infinity]and the category Repab(Se). This duality provides an unexpected tensor structure on the category Op[infinity]/t C[infinity].by Inna Entova Aizenbud.Ph. D
Deligne-Knop tensor categories and functoriality
Get PDFA general construction of Knop creates a symmetric monoidal category from any regular category and a fixed degree function . A special case of this construction are the Deligne categories and . We discuss when a functor between regular categories induces a symmetric monoidal functor . We then give a criterion when a pair of adjoint functors between two regular categories lifts to a pair of adjoint functors between and
Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra
Get PDFIn this paper, we study the representations of the periplectic Lie
superalgebra using the Duflo-Serganova functor. Given a simple
-module and a certain element of
rank , we give an explicit description of the composition factors of the
-module , which is defined as the homology of the
complex In particular, we
show that this -module is multiplicity-free. We then use
this result to give a simple explicit combinatorial formula for the
superdimension of a simple integrable finite-dimensional
-module, based on its highest weight. In particular, this
reproves the Kac-Wakimoto conjecture for , which was proved
earlier by the authors.Comment: ver 2: proof of prop. 3.2.2 significantly shortene
McKay trees
Get PDFGiven a finite group and its representation , the corresponding
McKay graph is a graph whose vertices are the irreducible
representations of ; the number of edges between two vertices of
is . The collection of all
McKay graphs for a given group encodes, in a sense, its character table.
Such graphs were also used by McKay to provide a bijection between the finite
subgroups of and the affine Dynkin diagrams of types , the
bijection given by considering the appropriate McKay graphs.
In this paper, we classify all (undirected) trees which are McKay graphs of
finite groups and describe the corresponding pairs ; this
classification turns out to be very concise.
Moreover, we give a partial classification of McKay graphs which are forests,
and construct some non-trivial examples of such forests.Comment: Ver2: Theorem A and its proof corrected, many examples adde
Abelian envelopes of rigid symmetric monoidal categories
No full textI will define what is an abelian envelope of a fixed rigid symmetric monoidal category, in the sense of Deligne. I will also give several examples and applications, both in characteristic zero and in positive characteristic.Non UBCUnreviewedAuthor affiliation: Ben Gurion UniversityResearche
Monoidal abelian envelopes and a conjecture of Benson--Etingof
Get PDFWe give several criteria to decide whether a given tensor category is the
abelian envelope of a fixed symmetric monoidal category. As a main result we
prove that the category of finite-dimensional representations of a semisimple
simply connected algebraic group is the abelian envelope of the category of
tilting modules. Benson and Etingof conjectured that a certain limit of finite
symmetric tensor categories is tensor equivalent to the finite dimensional
representations of in characteristic . We use our results on the
abelian envelopes to prove this conjecture and its variants for any prime .Comment: ver 3: Generalized the results to arbitrary primes p (ver 1,2 dealt
with the case p=2), ver 2: minor fix in the definition of abelian envelop
Deligne categories and the limit of categories
Get PDFFor each integer a tensor category is constructed, such that exact
tensor functors classify dualizable -dimensional
objects in not annihilated by any Schur functor. This means that is
the "abelian envelope" of the Deligne category . Any tensor functor
is proved to factor either through or
through one of the classical categories with . The
universal property of implies that it is equivalent to the categories
, (,
not integer) suggested by Deligne as candidates for the role of abelian
envelope.Comment: v3: lemma added to section 9, v4: some typos fixed, v5: fixed support
acknowledgement, v6: minor fix in definition of abelian envelope, v7: fixed
typo in preliminarie
Semisimplification of the category of tilting modules for GL_n
Get PDFWe describe the semisimplification of the monoidal category of tilting
modules for the algebraic group GL_n in characteristic p > 0. In particular, we
compute the dimensions of the indecomposable tilting modules modulo p.Comment: This version corrects a minor mistake in the proof of Lemma 3.4 in
the published versio
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