Weyl-Kac Character Formula Research Articles

The Ariki–Koike algebras and Rogers–Ramanujan type partitions

Ariki and Mathas (Math Z 233(3):601–623, 2000) showed that the simple modules of the Ariki–Koike algebras HC,v;Q1,…,Qm(G(m,1,n))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}_{\\mathbb {C},v;Q_1,\\ldots , Q_m}\\big (G(m, 1, n)\\big )$$\\end{document} (when the parameters are roots of unity and v≠1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v \ e 1$$\\end{document}) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit their generating function relation for the v=-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v=-1$$\\end{document} case. In particular, this v=-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v=-1$$\\end{document} scenario is of special interest as the corresponding Kleshchev multipartitions are closely tied with generalized Rogers–Ramanujan type partitions when Q1=⋯=Qa=-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_1=\\cdots =Q_a=-1$$\\end{document} and Qa+1=⋯=Qm=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_{a+1}=\\cdots =Q_m =1$$\\end{document}. Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for the above choice of parameters. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block, which are, as widely known, labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities for the m=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=2$$\\end{document} scenario.

The modular Weyl–Kac character formula

We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated one degree. Each of these homogeneous representations is one-dimensional and can be cohomologically constructed via a BGG resolution involving every (infinite dimensional) standard representation of the category. We hence determine the complete first row of the inverse parabolic p-Kazhdan–Lusztig matrix for an arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises the Weyl–Kac character formula to all Coxeter systems (and their parabolics) and proves that this generalised formula is rigid with respect to base change to fields of arbitrary characteristic.

Weyl-Kac character formula for affine Lie algebras in Deligne's categories

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L(X,k) of the parabolic Verma module M(X,k) of level k, where X is an indecomposable object of Deligne's category Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions that the highest weight of X plus the level gives a fundamental weight, t is transcendental, and the base field k has characteristic 0. We compare our result to the partial result in [7, Problem 6.2], and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula, [12, Formula (6.12)].

Weyl–Kac type character formula for admissible representations of Borcherds–Kac–Moody Lie superalgebras

We prove that Borcherds–Kac–Moody Lie superalgebras yield Weyl–Kac type character formulas for some specific types of admissible representations. Our results generalize the results of Kac and Wakimoto (Proc Natl Acad Sci USA 85:4956–4960, 1988) by extending the Weyl–Kac character formula for integrable irreducible highest weight modules over Kac–Moody algebras to the case of admissible irreducible highest weight modules over Borcherds–Kac–Moody Lie superalgebras.

Characters of Integrable Highest Weight Modules over a Quantum Group

We show that the Weyl-Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac-Moody Lie algebra, when the parameter $q$ is not a root of unity.

Hall–Littlewood polynomials and characters of affine Lie algebras

The Weyl–Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac–Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall–Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras Cn(1), A2n(2) and Dn+1(2). Through specialisation this yields generalisations for Bn(1), Cn(1), A2n−1(2), A2n(2) and Dn+1(2) of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers–Ramanujan, Andrews–Gordon and Göllnitz–Gordon q-series as special, low-rank cases.

The elliptic Weyl character formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].

Dirac Induction for Loop Groups

Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.

Kac’s conjecture from Nakajima quiver varieties

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.

Progress on Numerator Expansions for Affine Kac-Moody Algebras

The Weyl-Kac character formula for affine Kac-Moody algebras is recast as a quotient whose numerator and denominator can both be expressed as infinite sums of characters of irreducible highest weight representations of simple Lie subalgebra of the same rank. The denominator expansions, which coincide with well known Macdonald identities, are expressed here in terms of infinite series of characters, specified by particular types of partitions, subject to rank-dependent modification rules. It is shown that certain numberings of the associated Young diagrams provide a convenient framework for writing down contributions to the corresponding numerator expansions. In the case of the seven infinite series of affine Kac-Moody algebras that are indexed by their rank, progress is reported on the extent to which their numerator expansions can be completely determined.

A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard [Formula: see text]-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard [Formula: see text]-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic [Formula: see text]-module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilson's approach for affine Lie algebras of higher ranks, say for [Formula: see text], n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.

Multiplets of representations and Kostant's Dirac operator for equal rank loop groups

Let g be a semi-simple Lie algebra and let h be a reductive subalgebra of maximal rank in g. Given any irreducible representation of g, consider its tensor product with the spin representation associated to the orthogonal complement of h in g. Gross, Kostant, Ramond, and Sternberg recently proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h, called a multiplet. Kostant then constructed a formal h-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of h-representations corresponding to the given representation of g. We reproduce these results in the Kac-Moody setting for the extended loop algebras Lg and Lh. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of Lh associated to any irreducible positive energy representation of Lg. We construct a Lh-equivariant operator, analogous to Kostant's Dirac operator, on the tensor product of a representation of Lg with the spin representation associated to the complement of Lh in Lg. We then prove that the kernel of this operator gives the Lh-multiplet corresponding to the original representation of Lg.

Multiplets of representations and Kostant's Dirac operator for equal rank loop groups

Let $\mathfrak {g}$ be a semisimple Lie algebra, and let $\mathfrak {h}$ be a reductive subalgebra of maximal rank in $\mathfrak {g}$. Given any irreducible representation of $\mathfrak {g}$, consider its tensor product with the spin representation associated to the orthogonal complement of $\mathfrak {h}$ in $\mathfrak {g}$. Recently, B. Gross, B. Kostant, P. Ramond, and S. Sternberg [2] proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of $\mathfrak {h}$, called a multiplet. Kostant [7] then constructed a formal $\mathfrak {h}$-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of $\mathfrak {h}$-representations corresponding to the given representation of $\mathfrak {g}$. We reproduce these results in the Kac-Moody setting for the extended loop algebras $\tilde {L}\mathfrak {g}$ and $\tilde {L}\mathfrak {h}$. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of $L\mathfrak {h}$ associated to any irreducible positive energy representation of $L\mathfrak {g}$. We construct an $L\mathfrak {h}$-equivariant operator, analogous to Kostant's Dirac operator, on the tensor product of a representation of $L\mathfrak {g}$ with the spin representation associated to the complement of $L\mathfrak {h}$ in $L\mathfrak {g}$. We then prove that the kernel of this operator gives the $L\mathfrak {h}$-multiplet corresponding to the original representation of $L\mathfrak {g}$.

Character formula of Kac–Wakimoto type for generalized Kac–Moody algebras

We prove a character formula of Kac–Wakimoto type for generalized Kac–Moody algebras. A character formula of this type is a generalization of the Weyl–Kac character formula, and is proved by Kac–Wakimoto in the case of Kac–Moody algebras. We remark that the formula is a generalization of that of Kac–Wakimoto even in the case of Kac–Moody algebras of indefinite type.

Realization of level one representations of U q(g) at a root of unity

Using vertex operators, we construct explicitly Lusztig's ℤ[q,q−4]-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the character is given by the Weyl-Kac character formula.

Elements of the annihilating ideal of a standard module

The construction of affine Lie algebras by means of vertex operators has proved to be a remarkably fruitful line of research by motivating, being motivated by, and linking together a disparate range of sectors of mathematics and physics. Vertex operators entered mathematics through the work of J. Lepowsky and R. L. Wilson who gave an explicit construction of the affne algebra A’,l’ [LWl]. Their vertex operators were similar to ones used by physicists in the string model. Later on, I. Frenkel, V. Kac [FK], and G. Segal [S], found representations of afhne algebras using “untwisted” vertex operators. A large amount of very interesting research has followed stimulated by this observation and by the connection between classical combinatorial identities and the Weyl-Kac character formula [K, LM, LPl-2, LW2-4, Ma, Mil-21. A deeper understanding of vertex operators has been reached more recently as a result of work by Borcherds [B] who investigated general vertex operator algebras, and by Frenkel, Lepowsky, and A. Meurman in a series of papers, culminating in the recent book [FLM]. In particular, these three authors showed that the Monster finite simple group is the automorphism group of a certain vertex operator algebra, thus giving a natural realization of this largest among sporadic groups. This also established a surprising connection between the Monster and two-dimensional conformal quantum field theory, which is the physicists’ analogue of the theory of vertex operator algebras; cf. [BPZ, DL, FHL, FJ, TK, T, ZF, FLM].

A recursive formula for characters of simple Lie algebras

In 1964, Antoine and Speiser published succinct and elegant formulae for the characters of the irreducible highest weight modules for the Lie algebras A 2 and B 2. (The result for A 2 may have been known as early as 1957.) These can be derived from a recursive formula for characters valid for all simple complex Kac-Moody Lie algebras for which the Weyl-Kac character formula holds.

Relative version of Weyl-Kac character formula

Relative version of Weyl-Kac character formula

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA 1 (1) .

New Weyl groups forA 1 (1) and characters of singular highest weight modules

We consider singular Verma modules overA 1 (1) , i.e., Verma modules for which the central charge is equal to minus the dual Coxeter number. We calculate the characters of certain factor modules of these Verma modules. In one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights. We introduce new Weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves. Using these Weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semisimple Lie algebras (respectively Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras) so that the new Weyl groups play the role of the usual Weyl group (respectively affine Weyl group).