Starting from Jacobi-Trudi's type determinental expressions for the Schur functions corresponding to types $B,C$ and $D,$ we define a natural $q$-analogue of the multiplicity $[V(λ):M(μ)]$ when $M(μ)$ is a tensor product of row or column shaped modules defined by $μ$. We prove that these $q$-multiplicities are equal to certain Kostka-Foulkes polynomials related to the root systems $C$ or $D$. Finally we derive formulas expressing the associated multiplicities in terms of Kostka numbers.
Using the twisted denominator identity, we derive a closed form root multiplicity formula for all symmetrizable Borcherds-Bozec algebras and discuss its applications including the case of Monster Borcherds-Bozec algebra. In the second half of the paper, we provide the Schofield constuction of symmetric Borcherds-Bozec algebras.
We study a correction factor for Kac-Moody root systems which arises in the theory of $p$-adic Kac-Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of the correction factor can be encoded as a collection of polynomials $m_λ\in \mathbb{Z}[t]$ indexed by positive imaginary roots $λ$. At $t=0$ these polynomials evaluate to the root multiplicities, so we consider $m_λ$ to be a $t$-deformation of $\mathrm{mult} (λ)$. We generalize the Peterson algorithm and the Berman-Moody formula for root multiplici…
As a first application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra $\,\mathfrak{g}\,$ with a non-simply laced Dynkin diagram. We prove that in type $\,B, C\,$ or $\,F\,$ these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of $\,\mathfrak{g}.\,$
We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, using the combinatorics of Lyndon words we construct a basis for the root spaces correspondin…
In this paper, we calculate the dimension of root spaces $\mathfrak{g}_λ$ of a special type rank $3$ Kac-Moody algebras $\mathfrak{g}$. We first introduce a special type of elements in $\mathfrak{g}$, which we call elements in standard form. Then, we prove that any root space is spanned by these elements. By calculating the number of linearly independent elements in standard form, we obtain a formula for the dimension of root spaces $\mathfrak{g}_λ$, which depends on the root $λ$.
We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. The framework is general, but we only work out specifics in rank two. In that case we give explicit bounds. These turn out to be quite accurate, and in many cases exact, even for some fairly large roots.
We give a representation-theoretic interpretation of the Langlands character duality of Frenkel and Hernandez, and show that the "Langlands branching multiplicities" for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.
We prove that the weight multiplicities of the integrable irreducible highest weight module over the Kac-Moody algebra associated to a quiver are equal to the root multiplicities of the Kac-Moody algebra associated to some enlarged quiver. To do this, we use the Kac conjecture for indivisible roots and a relation between the Poincare polynomials of quiver varieties and the Kac polynomials, counting the number of absolutely irreducible representations of the quiver over finite fields. As a corollary of this relation, we get an explicit formula for the Poincare polynomials of quiver varieties, w…
We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann's Path Model, and geometrically, using isogenies between algebaric groups defined over an algebraically closed field of positive characteristic.
We prove the Plancherel formula for hypergeometric functions associated to a root system in the situation when the root multiplicities are negative (but close to 0). As a result we obtain a classification of the hypergeometric functions that are square integrable, and we find a closed formula for their square norm as a function of the root multiplicities.
For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{α_i \mid 0 \leq i \leq n-1\}$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module with highest weight $kΛ_0$. It is known that there are finitely many maximal dominant weights of $V(kΛ_0)$. Using the crystal base realization of $V(kΛ_0)$ and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form $kΛ_0 - λ^\ell_{a,b}$ where $ λ^\ell_{a,b} = \ellα_0 + (\ell-b)α_1 + (\ell-(b+1))α_2 + \cdots + α…
Consider the affine Lie algebra $\hat{s\ell}(n)$ with null root $δ$, weight lattice $P$ and set of dominant weights $P^+$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\hat{s\ell}(n)$-module with level $k \geq 1$ highest weight $kΛ_0$. Let $wt(V)$ denote the set of weights of $V(kΛ_0)$. A weight $μ\in wt(V)$ is a maximal weight if $μ+ δ\not\in wt(V)$. Let $max^+(kΛ_0)= max(kΛ_0)\cap P^+$ denote the set of maximal dominant weights which is known to be a finite set. In 2014, the authors gave the complete description of the set $max^+(kΛ_0)$. In subsequent pap…
We prove an analogue of Koszul duality for category $\mathcal{O}$ of a reductive group $G$ in positive characteristic $\ell$ larger than 1 plus the number of roots of $G$. However there are no Koszul rings, and we do not prove an analogue of the Kazhdan--Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus 2.
The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …
We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.
We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups. As an application of our formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.
Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostant's partition function. For $ξ$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(ξ)=\sum c_i q^i$ where $c_i$ is the number of ways $ξ$ can be written as a sum of $i$ positive roots of $\mathfrak{g}$. In this way, the evaluation of Kostant's weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight represen…
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$ which act locally finitely on $M$. A subalgebra $\mathfrak{l}\subset\mathfrak{g}$ for which there exists an irreducible module $M$ with $\mathfrak{g}[M]=\mathfrak{l}$ is called a Fernando-Kac subalgebra of $\mathfrak{g}$. A Fernando-Kac subalgebra of $\mathfrak{g}$ is of finite type if in addition $M$ can be chosen to have finite Jordan-Hölder $\mathfrak{l}…
In this paper, we study the Euler transform on linear ordinary differential operators on $\mathbb{P}^{1}$. The spectral type is the tuple of integers which count the multiplicities of local formal solutions with the same leading terms. We compute the changes of spectral types under the action of the Euler transform and show that the changes of spectral types generate a transformation group of a $\mathbb{Z}$-lattice which is isomorphic to a quotient lattice of a Kac-Moody root lattice with the Weyl group as the transformation group.
We first provide an explicit combinatorial description of the Auslander-Reiten quiver $Γ^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q'(D^{(i)}_{n+1})$ $(i=1,2)$ and the quiver Hecke algebra $R_{D_{n+1}}$ associated to $D_{n+1}$ $(n \ge 3)$, by using the combinatorial description and the generalized quantum affine Schur-Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category $\Rep(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roo…
We exhibit for all positive integers r, an explicit cellular structure for the endomorphism algebra of the r'th tensor power of an integral form of the Weyl module with highest weight d of the quantised enveloping algebra of sl2. When q is specialised to a root of unity of order bigger than d, we consider the corresponding specialisation of the tensor power. We prove one general result which gives sufficient conditions for the commutativity of specialisation with the taking of endomorphism algebras, and another which relates the multiplicities of indecomposable summands to the dimensions of si…
The natural representation of the quantized affine algebra of type A can be defined via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for a type A quantum group at a root of unity to the action of the quantized affine algebra as the rank tends towards infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional modules for the corresponding quatum group at a root of unity into Fock spaces of different charges and define an action of an affine quantum symm…
We give an equivariant version of the Saito duality which can be regarded as a Fourier transformation on Burnside rings. We show that (appropriately defined) reduced equivariant monodromy zeta functions of Berglund-Hübsch dual invertible polynomials are Saito dual to each other with respect to their groups of diagonal symmetries. Moreover we show that the relation between "geometric roots" of the monodromy zeta functions for some pairs of Berglund-Hübsch dual invertible polynomials described in a previous paper is a particular case of this duality.
For a Dynkin quiver $Q$ of type ADE and a sum $β$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $β$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl d…
