scholarly journals Integral metaplectic modular categories

2020 ◽  
Vol 29 (05) ◽  
pp. 2050032
Author(s):  
Adam Deaton ◽  
Paul Gustafson ◽  
Leslie Mavrakis ◽  
Eric C. Rowell ◽  
Sasha Poltoratski ◽  
...  
Keyword(s):  
Finite Group ◽  
Braid Group ◽  
Fusion Category ◽  
Group Representations ◽  
Link Invariant ◽  
Fusion Rules ◽  
Kauffman Polynomial ◽  
A Finite Group ◽  
Special Case ◽  
Braided Fusion Category

A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories [Formula: see text] with the fusion rules of [Formula: see text] we determine the finite group [Formula: see text] for which [Formula: see text] is braided equivalent to [Formula: see text]. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

scholarly journals Mapping class group representations from Drinfeld doubles of finite groups

2020 ◽  
Vol 29 (05) ◽  
pp. 2050033
Author(s):  
Jens Fjelstad ◽  
Jürgen Fuchs
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Group Representations ◽  
Torelli Group ◽  
Group Data ◽  
A Finite Group ◽  
Marked Points

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group [Formula: see text], focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular, we show that they have finite images, and that for surfaces of genus at least [Formula: see text] their restriction to the Torelli group is non-trivial if and only if [Formula: see text] is non-abelian.

scholarly journals BRAID GROUP REPRESENTATIONS ARISING FROM THE YANG–BAXTER EQUATION

2010 ◽  
Vol 19 (04) ◽  
pp. 525-538 ◽  
Author(s):  
JENNIFER M. FRANKO
Keyword(s):  
Group Algebra ◽  
Braid Group ◽  
Roots Of Unity ◽  
Baxter Equation ◽  
Group Representations ◽  
Link Invariant

This paper aims to determine the images of the braid group under representations afforded by the Yang–Baxter equation when the solution is a non-trivial 4 × 4 matrix. Making the assumption that all the eigenvalues of the Yang–Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant.

J-Equivalence of group representations

1971 ◽  
Vol 70 (1) ◽  
pp. 9-14 ◽  
Author(s):  
V. P. Snaith
Keyword(s):  
Finite Group ◽  
Galois Group ◽  
Complex Representation ◽  
Group Representations ◽  
Root Of Unity ◽  
Representation Ring ◽  
A Finite Group

If G is a finite group of order N and ΓN is the Galois group of Q(w) over Q, where w is a primitive Nth root of unity then ΓN acts on the complex representation ring, R(G), of G. The group of co-invariants is denoted by R(G)ΓN = R(G)/W(G).

Classification of Finite Group-Frames and Super-Frames

2007 ◽  
Vol 50 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Deguang Han
Keyword(s):  
Finite Group ◽  
Equivalence Classes ◽  
Group Representations ◽  
Exact Number ◽  
A Finite Group ◽  
Frame Representations

AbstractGiven a finite group G, we examine the classification of all frame representations of G and the classification of all G-frames, i.e., frames induced by group representations of G. We show that the exact number of equivalence classes of G-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number L such that there exists an L-tuple of strongly disjoint G-frames.

Varieties and modules with vanishing cohomology

1994 ◽  
Vol 116 (2) ◽  
pp. 245-251 ◽  
Author(s):  
Jon F. Carlson ◽  
Geoffrey R. Robinson
Keyword(s):  
Finite Group ◽  
Projective Module ◽  
Group Algebras ◽  
Finite Characteristic ◽  
Principal Block ◽  
Vanishing Of Cohomology ◽  
A Finite Group ◽  
Special Case

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context of kG-modules where k is a field of finite characteristic p and G is a finite group whose order is divisible by p. Aside from some general considerations, the main results of [2] related the existence of kG-modules M with H*(G, M) = 0 to the structure of the centralizers of the p-elements in G. Specifically it was shown that there exists a non-projective module M in the principal block of kG with H*(G, M) = 0 whenever the centralizer of some p-element of G is not p-nilpotent. The converse was proved in the special case that the prime p is an odd integer (p > 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.

A pair of characteristic subgroups for pushing-up. II

2012 ◽  
Vol 56 (1) ◽  
pp. 71-133 ◽  
Author(s):  
George Glauberman
Keyword(s):  
Finite Group ◽  
Local Analysis ◽  
Nilpotence Class ◽  
Similar Theorem ◽  
A Finite Group ◽  
Special Case

AbstractMany problems about local analysis in a finite group G reduce to a special case in which G has a large normal p-subgroup satisfying several restrictions. In 1983, R. Niles and G. Glauberman showed that every finite p-group S of nilpotence class at least 4 must have two characteristic subgroups S1 and S2 such that, whenever S is a Sylow p-subgroup of a group G as above, S1 or S2 is normal in G. In this paper, we prove a similar theorem with a more explicit choice of S1 and S2.

On The Number of Classes of a Finite Group Invariant for Certain Substitutions

1974 ◽  
Vol 26 (5) ◽  
pp. 1090-1097 ◽  
Author(s):  
A. J. van Zanten ◽  
E. de Vries
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Symmetric Group ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Group Invariant ◽  
A Finite Group ◽  
Number Of Classes ◽  
Special Case

In this paper we consider representations of groups over the field of the complex numbers.The nth-Kronecker power σ⊗n of an irreducible representation σ of a group can be decomposed into the constituents of definite symmetry with respect to the symmetric group Sn. In the special case of the general linear group GL(N) in N dimensions the decomposition of the defining representation at once provides irreducible representations of GL(N) [9; 10; 11].

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