This article introduces a new term "splint" and classifies the splints of the classical root systems. The motivation comes from representation theory of semisimple Lie algebras. In a few instances, splints play a role in determining branching rules of a module over a complex semisimple Lie algebra when restricted to a subalgebra. In these particular cases, the set of submodules with respect to the subalgebra themselves may be regarded as the character of another Lie algebra.
Splint is a decomposition of root system into union of root systems. Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We consider special embedding of Lie subalgebra to Lie algebra. We classify projections of algebra root systems using extended Dynkin diagrams and single out the conditions of splint appearance and coincidence of branching coefficients with weight multiplicities. While such a coincidence is not very common it is connected with Gelfand-Tsetlin basis.
Splints of root system of simple lie algebras appears naturally on studies of embedding of reductive subalgebras. A splint can be used to construct a branching rules as implementation of this idea simplifies calculation of branching coefficient. We extend the concept of splints to classical lie superalgebras cases as these algebras have wide application in physics.
Splint of root system of simple Lie algebra appears naturally in the study of (regular) embeddings of reductive subalgebras. It can be used to derive branching rules. Application of splint properties drastically simplifies calculations of branching coefficients. We study affine extension of splint root system of simple Lie algebra and obtain relations on theta and branching functions.
A root system is splint if it is a decomposition into a union of two root systems. Examples of such root systems arise naturally in studying embeddings of reductive Lie subalgebras into simple Lie algebras. Given a splint root system, one can try to understand its branching rule. In this paper we discuss methods to understand such branching rules, and give precise formulas for specific cases, including the restriction functor from the exceptional Lie algebra $\mathfrak{g}_2$ to $\mathfrak{sl}_3$.
Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We demonstrate that splint properties implementation drastically simplify calculations of branching coefficients.
Properly specializing the parameters in ``Schnizer modules'', for type A,B,C and D, we get its unique primitive vector. Then we show that the module generated by the primitive vector is an irreducible highest weight module of finite dimensional classical quantum groups at roots of unity.
Let $Φ$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $Φ$, $U$ be its maximal unipotent subgroup and $\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $Φ$ and $Ω$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\mathfrak{u}$ at the canonical form on $Ω$. We also find the dimension of $Ω$ in terms of the Weyl group of $Φ$. As a corollary, we determine all possible dimensions of irreducible complex represenations of the group $U$ for the c…
This paper presents an algorithm to compute the value of the inverse Laplace transforms of rational functions with poles on arrangements of hyperplanes. As an application, we present an efficient computation of the partition function for classical root systems.
Kostant's partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra. For a given weight the $q$-analog of Kostant's partition function is a polynomial where the coefficient of $q^k$ is the number of ways the weight can be written as a nonnegative integral sum of exactly $k$ positive roots. In this paper we determine generating functions for the $q$-analog of Kostant's partition function when the weight in question is the highest root of the classical Lie algebras of types $B$, $C$ and $D$.
We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenspaces appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.
We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.
This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the affine classical root systems and some claims for arbitrary (reduced) affine root systems. It continues our previous paper devoted to the non-affine case; interestingly, the affine theory clarifies the classification in the non-affine case. The lambda-sequences are associated with reduced decompositions (words) in affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners in the theory of irreducible repre…
The study of representations of affine Hecke algebras has led to a new notion of shapes and standard Young tableaux which works for the root system of any finite Coxeter group. This paper is completely independent of affine Hecke algebra theory and is purely combinatorial. We define generalized shapes and standard Young tableaux and show that these new objects coincide with the classical ones for root systems of Type A. The classical notions of conjugation of shapes, ribbon shapes, axial distances, and the row reading and column reading standard tableaux, have natural generalizations to the ro…
Noncompact forms of the Drinfeld-Jimbo quantum groups U_q(g) with (H_i)* = H_i, (X_i^{+-})* = s_i X_i^{-+} for s_i= +-1 are studied at roots of unity. This covers g = so(n,2p), su(n,p), so*(2l), sp(n,p), sp(l,R), and exceptional cases. Finite-dimensional unitary representations are found for all these forms, for even roots of unity. Their classical symmetry induced by the Frobenius-map is determined, and the meaning of the extra quasi-classical generators appearing at even roots of unity is clarified. The unitary highest weight modules of the classical case are recovered in the limit q -> 1.
We explicitly describe the defining relations for simple Lie algebra of vector fields with polynomial coefficients and its subalgebras of divergence free, hamiltonian and contact vector fields, and for the Poisson algebra (realized on polynomials). We consider generators of these Lie algebras corresponding to the systems of simple roots associated with the standard grading of these algebras. (These systems of simple roots are distinguished in the sense of Penkov and Serganova.
The normalizer $N_G(H_G)$ of a maximal torus $H_G$ in a semisimple complex Lie group $G$ does not in general allow a presentation as a semidirect product of $H_G$ and the corresponding Weyl group $W_G$. Meanwhile, splitting holds for classical groups corresponding to the root systems $A_\ell$, $B_\ell$, $D_\ell$. For the remaining classical groups corresponding to the root systems $C_\ell$ there still exists an embedding of the Tits extension of $W_G$ into normalizer $N_G(H_G)$. We provide explicit unified construction of the lifts of the Weyl groups into normalizers of maximal tori for classi…
A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems, we define a finite root system $R$ to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to $R$ a semisimple Lie algebra $L(R)$ together with a quasi-good grading on it. Thus one can construct nice basis of $L(R)$ by means of finite root systems. We classify finite maximal abelian subgroups $T$ in $\Aut(L…
For a fixed integer $t \geq 2$, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely $\text{GL}_{tn}, \text{SO}_{2tn+1}, \text{Sp}_{2tn}$ and $\text{O}_{2tn}$, evaluated at elements $ω^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, where $ω$ is a primitive $t$'th root of unity. The case of $\text{GL}_{tn}$ was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. Math., 2016). In this article, we give a uniform approach for all cases. In this article, we give a uniform approach for all cases…
In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our construction including the construction of a sort of Gelfand Zetlin basis for a generic irreducible representation of quantum GL(n) at odd roots of unity.
We construct a divided difference operator using GKM theory. This generalizes the classical divided difference operator for the cohomology of the complete flag variety. This construction proves a special case of a recent conjecture of Shareshian and Wachs. Our methods are entirely combinatorial and algebraic, and rely heavily on the combinatorics of root systems and Bruhat order.
Fix natural numbers $n \geq 1$, $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $ω$. In previous work with A. Ayyer (J. Alg., 2022), we studied the factorization of specialized irreducible characters of $\text{GL}_{tn}$, $\text{SO}_{2tn+1},$ $\text{Sp}_{2tn}$ and $\text{O}_{2tn}$ evaluated at elements to $ω^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$. In this work, we extend the results to the groups $\text{GL}_{tn+m}$ $(0 \leq m \leq t-1)$, $\text{SO}_{2tn+3}$, $\text{Sp}_{2tn+2}$ and $\text{O}_{2tn+2}$ evaluated at similar specializations: (1) for the $\text{GL}_{tn+m}(\mat…
We introduce and study the class of groups graded by root systems. We prove that if Φ is an irreducible classical root system of rank at least 2 and G is a group graded by Φ, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem we prove that for any reduced irreducible classical root system Φ of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_Φ(R) and the elementary Chevalley group E_Φ(R) have property (T). We also show that there exists a group with pro…
Let $M(\gl)$ be a Verma module for a basic classical simple Lie superalgebra $\fg \neq G(3)$ defined using the distinguished Borel subalgebra, and let $\gc$ be an isotropic positive root of $\fg.$ As a special case of our first main result we show that if $μ, \gl \in \fh^*$ with $\gl-μ= \gc$ we have $$\dim \Hom_{\sfg}(M(μ),M(\gl))\le 1.$$ This result applies to the construction of Šapovalov elements for isotropic roots. The proof rests on a comparison with the corresponding result for a certain simple Lie algebra $G$.
We study root-theoretic Young diagrams to investigate the existence of a Lie-type uniform and nonnegative combinatorial rule for Schubert calculus. We provide formulas for (co)adjoint varieties of classical Lie type. This is a simplest case after the (co)minuscule family (where a rule has been proved by H.Thomas and the second author using work of R.Proctor). Our results build on earlier Pieri-type rules of P.Pragacz-J.Ratajski and of A.Buch-A.Kresch-H.Tamvakis. Specifically, our formula for OG(2,2n) is the first complete rule for a case where diagrams are non-planar. Yet the formulas possess …
