Characters Of Irreducible Representations Research Articles

Kac-Moody and Virasoro characters from the perturbative Chern-Simons path integral

We evaluate to one loop the functional integral that computes the partition functions of Chern-Simons theories based on compact groups, using the background field method and a covariant gauge fixing. We compare our computation with the results of other, less direct methods. We find that our method correctly computes the characters of irreducible representations of Kac-Moody algebras. To extend the computation to non-compact groups we need to perform an appropriate analytic continuation of the partition function of the compact group. Non-vacuum characters are found by inserting a Wilson loop in the functional integral. We then extend our method to Euclidean Anti-de Sitter pure gravity in three dimensions. The explicit computation unveils several interesting features and lessons. The most important among them is that the very definition of gravity in the first-order Chern-Simons formalism requires non-trivial analytic continuations of the gauge fields outside their original domains of definition.

Quiver theories and formulae for nilpotent orbits of Exceptional algebras

We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.

On the expansion of certain vector-valued characters of $U_q (\mathfrak{gl}_n)$ with respect to the Gelfand–Tsetlin basis

Macdonald polynomials are an important class of symmetric functions, with connections to many different fields. Etingof and Kirillov showed an intimate connection between these functions and representation theory: they proved that Macdonald polynomials arise as (suitably normalized) vector-valued characters of irreducible representations of quantum groups. In this paper, we provide a branching rule for these characters. The coefficients are expressed in terms of skew Macdonald polynomials with plethystic substitutions. We use our branching rule to give an expansion of the characters with respect to the Gelfand-Tsetlin basis. Finally, we study in detail the $q=0$ case, where the coefficients factor nicely, and have an interpretation in terms of certain $p$-adic counts.

ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

Quiver theories for moduli spaces of classical group nilpotent orbits

We approach the topic of Classical group nilpotent orbits from the perspective of their moduli spaces, described in terms of Hilbert series and generating functions. We review the established Higgs and Coulomb branch quiver theory constructions for A series nilpotent orbits. We present systematic constructions for BCD series nilpotent orbits on the Higgs branches of quiver theories defined by canonical partitions; this paper collects earlier work into a systematic framework, filling in gaps and providing a complete treatment. We find new Coulomb branch constructions for above minimal nilpotent orbits, including some based upon twisted affine Dynkin diagrams. We also discuss aspects of 3d mirror symmetry between these Higgs and Coulomb branch constructions and explore dualities and other relationships, such as HyperKahler quotients, between quivers. We analyse all Classical group nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials.

Character varieties of double twist links

We compute both natural and smooth models for the $SL_2(\mathbb C)$ character varieties of the two component double twist links, an infinite family of two-bridge links indexed as $J(k,l)$. For each $J(k,l)$, the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form $C\times \mathbb C$ where $C$ is a curve. The same is true of the canonical component. We compute the genus of this curve, and the degree of irrationality of the canonical component. We realize the natural model of the canonical component of the $SL_2(\mathbb C)$ character variety of the $J(3,2m+1)$ links as the surface obtained from $\mathbb{P}^1\times \mathbb{P}^1$ as a series of blow-ups.

On the irreducibility of irreducible characters of simple Lie algebras

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For S L ( r ) SL(r) the irreducibility result is the following: let λ = ( a 1 ≥ a 2 ≥ ⋯ ≥ a r − 1 ≥ 0 ) \lambda =(a_1\geq a_2\geq \cdots \geq a_{r-1}\geq 0) be the highest weight of an irreducible rational representation V λ V_{\lambda } of S L ( r ) SL(r) . Assume that the integers a 1 + r − 1 , a 2 + r − 2 , ⋯ , a r − 1 + 1 a_1+r-1, ~a_2+r-2, \cdots , a_{r-1}+1 are relatively prime. Then the character χ λ \chi _{\lambda } of V λ V_{\lambda } is strongly irreducible in the following sense: for any natural number d d , the function χ λ ( g d ) , g ∈ S L ( r , C ) \chi _{\lambda }(g^d), ~g\in SL(r,\mathbb {C}) is irreducible in the ring of regular functions of S L ( r , C ) SL(r,\mathbb {C}) .

Factor groups, semidirect product and quantum chemistry

In this article, we prove some general theorems about representations of finite groups arising from the inner semidirect product of groups. We show how these results can be used for standard applications of group theory in quantum chemistry through the orthogonality relations for the characters of irreducible representations. In this context, conditions for transitions between energy levels, projection operators, and basis functions were determined. This approach applies to composite systems and it is illustrated by the dihedral group related to glycolate oxidase enzyme. © 2014 Wiley Periodicals, Inc.

General Linear and Functor Cohomology Over Finite Fields

In recent years, there has been considerable success in computing Extgroups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories [F-L-S], [F-S]. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p. The reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of Ext-groups. For example, the existence of nonzero Ext-groups of positive degree is equivalent to the existence of objects which are not “direct sums” of simple objects. Indeed, a knowledge of Ext-groups provides considerable knowledge of compound objects. In the study of modular representation theory of finite Chevalley groups such as GLn(Fq), Ext-groups play an even more central role: it has been shown in [CPS] that a knowledge of certain Ext-groups is sufficient to prove Lusztig’s Conjecture concerning the dimension and characters of irreducible representations. We consider two different categories of functors, the category F(Fq) of all functors from finite dimensional Fq-vector spaces to Fq-vector spaces, where Fq is the finite field of cardinality q, and the category P(Fq) of strict polynomial functors of finite degree as defined in [F-S]. The category P(Fq) presents several advantages over the category F(Fq) from the point of view of computing Extgroups. These are the accessibility of injectives and projectives, the existence of a base change, and an even easier access to Ext-groups of tensor products. This explains the usefulness of our comparison in Theorem 3.10 of Ext-groups in the category P(Fq) with Ext-groups in the category F(Fq). Weaker forms of this theorem have been known to us since 1995 and to S. Betley independently

Characters of An-1Hecke algebras at roots of unity

A Frobenius formula is presented for the characters, phi sigma p(q), of the irreducible representations, pi ( sigma ), of the An-1 Hecke algebras Hn(q) labelled by (m, k)-standard partitions, sigma , where q is a primitive pth root of unity with p=m+k. Using this result the characters phi sigma p(q) may be expressed as linear combinations of the characters, chi lambda p(q), of representations which are irreducible when q is not a root of unity. The appropriate linear combinations are found by using fusion modification rules. For 1<or=n<or=5 all the remaining characters of irreducible representations of Hn(q) are then found, allowing a complete tabulation to be made for 2<or=p<or=5, along with the corresponding decomposition matrices.

Characters of irreducible representations of the virasoro algebra

On obtient les caracteres des modules irreductibles non integrables de plus haut poids L ((h,0)). On determine les caracteres des familles L ((t,25)) et L((t,26)) duales de L ((h,1)) et de L ((h,0))

Characters of irreducible representations of the Lie algebra of vector fields on the circle

Characters of irreducible representations of the Lie algebra of vector fields on the circle

Characters of irreducible representations of real semisimple Lie groups

Characters of irreducible representations of real semisimple Lie groups

A property of the characters of irreducible representations of real semisimple Lie groups

A property of the characters of irreducible representations of real semisimple Lie groups