Chris Godsil, Karen Meagher
We determine all the multiplicity-free representations of the symmetric group. This project is motivated by a combinatorial problem involving systems of set-partitions with a specific pattern of intersection.
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We provide a classification of multiplicity-free inner tensor products of irreducible characters of symmetric groups, thus confirming a conjecture of Bessenrodt. Concurrently, we classify all multiplicity-free inner tensor products of skew characters of the symmetric groups. We also provide formulae for calculating the decomposition of these tensor products.
Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic. Remark: parallel work of Godsil and Meagher (arXiv…
Multiplicity-free induced characters of symmetric groups
本文结合代数与组合技术研究对称群子群及其不可约特征标的诱导特征标何时无重数，分类了 $n\geq 66$ 时的所有相关子群，并在 $n\geq 73$ 时确定了大多数群对应的不可约特征标。
In this paper we consider symmetric powers representation and exterior powers representation of finite groups, which generated by the representation which has finite dimension over the complex field. We calculate the multiplicity of irreducible component of two representations of some representation by using a character theory of representation and a pre-lambda-ring, for example, the regular representation.
We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are $GL_n$, and $SL_n$, characters of the skew Schur modules. Our result extends work of H. Thomas--A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.
We give a complete classification of the reductive symmetric pairs (G,H) for which the homogeneous space $(G \times H)/diag(H)$ is real spherical in the sense that a minimal parabolic subgroup has an open orbit. Combining with a criterion established in [T. Kobayashi--T. Oshima, Adv. Math. 2013], we give a necessary and sufficient condition for a reductive symmetric pair $(G,H)$ such that the multiplicities for the branching law of the restriction any admissible smooth representation of $G$ to $H$ have finiteness/boundedness property.
We show that the Kirillov--Reshetikhin crystal $B^{r,s}$ exists when $r$ is a node such that the Kirillov--Reshetikhin module $W^{r,s}$ has a multiplicity free classical decomposition.
We classify and construct all multiplicity-free plethystic products of Schur functions. We also compute many new (infinite) families of plethysm coefficients, with particular emphasis on those near maximal in the dominance ordering and those of small Durfee size.
Haglund's conjecture states that $\dfrac{\langle J_λ(q,q^k),s_μ\rangle}{(1-q)^{|λ|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $λ,μ$ and all non-negative integers $k$, where $J_λ$ is the integral form Macdonald symmetric function and $s_μ$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(λ,μ)$ satisfies $K_{λ,μ}=1$ or $K_{μ',λ'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.
Bounded multiplicity branching for symmetric pairs
本文研究单连通非紧半单李群在对称对上的表示分支，证明存在无限维不可约表示使其对所有对称对的限制均有有界重数，并刻画满足该性质的表示。
The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branching theorems that include the multiplicity-free property of some of known results such as the Clebsh--Gordan--Pieri formula for tensor products, the Plancherel theorem for Hermitian symmetric spaces (also for line bundle cases), the Hua--Kostant--Schmid $K$-type formula, and the …
Symmetric spaces and skew multiplicity free representations
本文给出非紧约化实李群抛物子群具有阿贝尔幂单根的等价条件，并利用 Clifford 代数与三次 Dirac 算子推导紧致对称空间的上同调公式。
We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.
Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains
本文研究拟对称Siegel域上两步幂零Lie群的无重数表示，给出了该性质成立的充要条件，并证明了不可约酉表示的不相交性、Bergman空间上酉表示的无重数性与群作用余迷向性的等价关系。
For any given bounded symmetric domain, we prove the existence of commutative $C^*$-algebras generated by Toeplitz operators acting on any weighted Bergman space. The symbols of the Toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain. These subgroups include the maximal compact groups of biholomorphisms. We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known mul…
In this paper we show that two skew diagrams lambda/mu and alpha/beta can represent the same multiplicity free skew character [lambda/mu]=[alpha/beta] only in the the trivial cases when lambda/mu and alpha/beta are the same up to translation or rotation or if lambda=alpha is a staircase partition lambda=(l,l-1,...,2,1) and lambda/mu and alpha/beta are conjugate of each other.
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e. skew Schur functions having Littlewood-Richardson coefficients always equal to $1$ over the full interval.
We study weight posets of weight multiplicity free (=wmf) representations $R$ of reductive Lie algebras. Specifically, we are interested in relations between $\dim R$ and the number of edges in the Hasse diagram of the corresponding weight poset, $# E(R)$. We compute the number of edges and upper covering polynomials for the weight posets of all wmf-representations. We also point out non-trivial isomorphisms between weight posets of different irreducible wmf-representations. Our main results concern wmf-representations associated with periodic gradings or Z-gradings of simple Lie algebras. For…
Multiplicity One and van Dijk's Conjecture for Connected Complex Symmetric Spaces
本文通过不变论方法证明三类经典剩余族的正则性，修复此前证明漏洞，结合相关结果确立连通复对称空间的重数1，并最终证明van Dijk猜想。
Let $V$ be a complex finite dimensional super vector space with an action of a connected semisimple group $G$. We classify those pairs $(G,V)$ for which all homogeneous components of the super symmetric algebra of $V$ decompose multiplicity-free.
Multiplicity free Weil representations arising from exceptional groups
该研究借助例外theta对应，证明p进群的部分Weil表示具有无重数性质，同时确定了这些表示的不可约商，推进了相关表示论研究。
Let $G$ be a connected reductive group defined over $\CC$ with a finite dimensional representation $V$. The action of $G$ is said to be skew multiplicity-free (SMF) if the exterior algebra $\bigwedge V$ contains no irreducible representation of $G$ with multiplicity $> 1$. In this paper we classify all such representations.
Let $(M,ω)$ be a connected symplectic manifold on which a connected Lie group $G$ acts properly and in a Hamiltonian fashion with moment map $μ:M \lra \mf g^*$. Our purpose is investigate multiplicity-free actions, giving criteria to decide a multiplicity freenes of the action. As an application we give the complete classification of multiplicity-free actions of compact Lie groups acting isometrically and in a Hamiltonian fashion on Hermitian symmetric spaces of noncompact type. Successively we make a connection between multiplicity-free actions on $M$ and multiplicity-free actions on the symp…
Multiplicity-free representations of certain nilpotent Lie groups over Siegel domains of the second kind
本文研究第二类Siegel域上广义Heisenberg群及其子群的全纯乘子表示，给出无重数性的充要条件，并将其与余迷向作用、可见作用及不变微分算子代数的交换性联系起来。
In this paper we classify the multiplicity-free skew characters of the symmetric group. Furthermore we show that the Schubert calculus is equivalent to that of skew characters in the following sense: If we decompose the product of two Schubert classes we get the same as if we decompose a skew character and replace the irreducible characters by Schubert classes of the `inverse' partitions (Theorem 4.2).
Multiplicity in restricting minimal representations
本文研究实约化李群极小表示在约化对称对下的限制问题，通过分析子群对小幂零轨道的作用，证明其重数有界。
In general it is a difficult problem to construct the lattice of submodules $L(M)$ of a given module $M$. In \cite{St} R. P. Stanley outlined a method for constucting a distributive lattice from a knowledge of its join irreducibles. However it is not an easy task to identify all join irreducible submodules of a given module. In the case of a multiplicity free module $M$ we present a modifiiction of Stanley's method based on the composition factors of $M$. As input we require a set of submodules $A_1,\ldots , A_n$ whose submodule lattices are known and which contain all composition factors of $…
Divergence-free Lie algebras (also known as the special Lie algebras of Cartan type) are Lie algebras of volume-preserving transformation groups. They are simple in generic case. Dokovic and Zhao found a certain graded generalization of them. In this paper, we classify all the irreducible and indecomposable multiplicity-free modules of the simple generalized divergence-free Lie algebras.
Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(π),ρ)<= 1, for any irreducible representations πof G and ρof M. To do that we adapt the classical method of Ge…
A (quasi-)Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify multiplicity free Hamiltonian actions for (twisted) loop groups or, equivalently, multiplicity free (twisted) quasi-Hamiltonian manifolds for simply connected compact Lie groups. As a result we recover old and find new examples of these structures.
We show that there is a reflection type bijection between the indecomposable summands of two multiplicity free tilting modules $X$ and $Y$. This bijection fixes the common indecomposable summands of $X$ and $Y$ and sends indecomposable projective (resp., injective) summands of exactly one module to non-projective (resp., non-injective) summands of the other. Moreover, this bijection interchanges the two possible non-isomorphic complements of an almost complete tilting module.
We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GL_n(q) acting on the tensor product of the symmetric algebra and the exterior algebra of the natural GL_n(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GL_n(q) in the tensor product of the symmetric algebra and the exterior algebra of V, twisted by an arbitrary determinant power.
On multiplicity-free weight modules over quantum affine algebras
本文旨在构造并研究量子仿射代数的无重数权模，通过引入关于可对称化广义 Cartan 矩阵的可移性条件，探讨了其在代数结构与无穷维模实现中的应用，并计算了相关最高 $\ell$-权。
In this paper we determine, under some mild restrictions, the lattice of submodules $\gL$ of a module $M$ all of whose composition factors have multiplicity one. Such a lattice is distributive, and hence determined by its poset of down-sets $P$. We define a directed Ext graph $\Ext_\gL$ of $\gL$ and show that if $\Ext_\gL$ is acyclic, then $\Ext_\gL$ determines $P$. The result applies to multiplicity free indecomposable modules for finite dimensional algebras with acyclic Ext graph. It also applies to some deformed Verma modules which arise in the Jantzen sum formula basic classical simple Lie…
Corrigendum to "Modular Gelfand pairs and multiplicity-free representations"
本文指出“模Gelfand对与无重表示”中部分无重性结果过于宽泛，给出反例和部分修正。
The multiplicity free spaces with a one dimensional quotient were introduced by Thierry Levasseur in [11]. Recently, the author has shown that the algebra of differential operators on such spaces which are invariant under the semi-simple part of the group is a Smith algebra ([17]). We give here the classification of these spaces which are indecomposable, up to geometric equivalence. We also investigate whether or not these spaces are regular or of parabolic type as a prehomogeneous vector space.
A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type respectively, and use them to construct multiplicity free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp. Jordan) algebras are simple. Three families of skew root systems of Lie type (resp. of Jordan type) are constructed and the corresponding Lie (resp. Jordan) algebras are iden…
Consider the restriction of an irreducible unitary representation $π$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ν$ occurring in the restriction $π|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $π$ and $ν$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results…
Construction of commuting difference operators for multiplicity free spaces
本文研究了满足特定公理的根系与主导权重基，构造了能同时对角化的差分算子，并推导出Pieri型规则，扩展了移位Jack多项式结果。
Linear groups of alternating type containing non-scalar elements with all but one eigenvalues of multiplicity 1
该研究确定了交错群、对称群及其泛中心扩张中，含除一个特征值外其余特征值重数均为1的非标量元的不可约表示，基域为任意特征的代数闭域。
In a recent paper, the authors have constructed a large class of operators in the Cowen-Douglas class Cowen-Douglas class of the unit disc $\mathbb D$ which are {\em homogeneous} with respect to the action of the group Möb -- the Möbius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb{D}$. The {\em associated representation} for each of these operators is {\em multiplicity free}. Here we give a different independent construction of all homogeneous operators in the Cowen-Douglas class with multiplicity free associated representation and verify that they are exactly the…
We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible $gl(m|n)$-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of $gl(m|n)$. In the former case we find as well explicit formulas for…
We compute the sheaf of automorphisms of a multiplicity free Hamiltonian manifold over its momentum polytope and show that its higher cohomology groups vanish. Together with a theorem of Losev, arXiv:math/0612561, this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determined by its momentum polytope and its principal isotropy group.
The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits
本文研究了有限维单李代数中与刚性幂零轨道关联的无重数本原理想的数量，求解了 ${\ m E}_8$ 型李代数的两个最大刚性轨道，并计算了特征 $p>5$ 的代数闭域上对应约化包络代数中小模的数量。
Let $K$ be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space $V$. One calls this a {\em multiplicity-free} action whenever the $K$-isotypic components of $\C[V]$ are $K$-irreducible. We have shown that this is the case if and only if the moment map $τ:V\rightarrow\k^*$ for the action is finite-to-one on $K$-orbits. This is equivalent to a result concerning \gp s associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere.
We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.
We prove that any finite W-algebra U(g,e) admits a one-dimensional representation fixed by the action of the component group of the centraliser of e. As a consequence, for any nilpotent orbit O in g there exists a multiplicity-free (and hence completely prime) primitive ideal of the universal enveloping algebra U(g) whose associated variety coincides with the Zariski closure of O.
Multiplicity-free tensor products of irreducible modules over simple algebraic groups in positive characteristic
该硕士论文针对正特征代数闭域上的单代数群，分类了$SL_2(k)$、$SL_3(k)$及$p=2$时$SL_n(k)$的单模无重数张量积，并给出$Sp_4(k)$的部分结果。
We prove a geometric criterion for the bounded multiplicity property of "small" infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full description of the triples $H \subset G \supset G'$ such that any irreducible admissible representations of $G$ with $H$-distinguished vectors have the bounded multiplicity property when restricted to the subgroup $G'$. This article also completes the proof of the general results announced in the previous paper [Adv. Math. 2021, Section 7].
We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of $G = \varinjlim G_n$ on a certain function space $\varinjlim L^2(G_n/K_n)$ is multiplicity free. That method is not applicable for direct limits of nonsymmetric …
We classify all momentum polytopes of rank one for multiplicity free quasi-Hamiltonian $K$-manifolds for simple and simply connected Lie groups $K$ by using the methods developed in a recent paper by F. Knop. This leads to lots of new concrete examples of multiplicity free quasi-Hamiltonian manifolds or equivalently, Hamiltonian loop group actions.
Let $(G,H)$ be a reductive spherical pair and $P\subset H$ a parabolic subgroup such that $(G,P)$ is spherical. The triples $(G,H,P)$ with this property are called multiplicity free systems and they are classified in this paper. Denote by $π^{H}_μ=\mathrm{ind}_{P}^{H}μ$ the Borel-Weil realization of the irreducible $H$-representation of highest weight $μ\in P^{+}_{H}$ and consider the induced representation $\mathrm{ind}_{P}^{G}χ_μ=\mathrm{ind}_{H}^{G}π^{H}_μ$, a multiplicity free induced representation. Some properties of the spectrum of the multiplicity free induced representations are discu…
We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dimensional cases in a systematic and synthetic manner. The key geometric condition in our theorem is an orbit-preserving anti-holomorphic diffeomorphism on the base space, which brings us to the concept of visible actions on complex manifolds.
Any multiplicity-free family of finite dimensional algebras has a canonical complete set of of pairwise orthogonal primitive idempotents in each level. We give various methods to compute these idempotents. In the case of symmetric group algebras over a field of characteristic zero, the set of canonical idempotents is precisely the set of seminormal idempotents constructed by Young. As an example, we calculate the canonical idempotents for semisimple Brauer algebras.
We determine the graded composition multiplicity in the symmetric algebra S(V) of the natural GL_n(q)-module V, or equivalently in the coinvariant algebra of V, for a large class of irreducible modules around the Steinberg module. This was built on a computation, via connections to algebraic groups, of the Steinberg module multiplicity in a tensor product of S(V) with other tensor spaces of fundamental weight modules.
We introduce a family of multivariate continuous-time pure birth and pure death chains, with birth and death rates defined in terms of the generalized binomial coefficients for multiplicity free actions. The state spaces for some of the introduced processes are some sets of partitions (equivalently, Young diagrams). The chains turn out to be the classical Markov processes obtained by restricting Biane's quantum Ornstein-Uhlenbeck semigroups to commutative C*-algebras related to Gelfand pairs built on Heisenberg groups.
We prove several results asserting that the action of a Banach-Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity free. These results require the existence of compatible antiholomorphic bundle maps and certain multiplicity freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,σ)$-weakly visible action (generalizing T.Koboyashi's visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for gro…
Multiplicity free and completely reducible tensor products for $\mathrm{SL}_3(\Bbbk)$ and $\mathrm{Sp}_4(\Bbbk)$
本文研究了正特征代数闭域上单代数群的两个简单模张量积何时无重数或完全可约的问题，开发了一般性判定工具，并对 $\mathrm{SL}_3(\Bbbk)$ 和 $\mathrm{Sp}_4(\Bbbk)$ 给出了完整解答。
Let $\g$ be a finite-dimensional complex semisimple Lie algebra and $\b$ a Borel subalgebra. Then $\g$ acts on its exterior algebra $\w\g$ naturally. We prove that the maximal eigenvalue of the Casimir operator on $\w\g$ is one third of the dimension of $\g$, that the maximal eigenvalue $m_i$ of the Casimir operator on $\w^i\g$ is increasing for $0\le i\le r$, where $r$ is the number of positive roots, and that the corresponding eigenspace $M_i$ is a multiplicity-free $\g$-module whose highest weight vectors corresponding to certain ad-nilpotent ideals of $\b$. We also obtain a result describi…
If $(G,V)$ is a multiplity free space with a one dimensional quotient we give generators and relations for the non-commutative algebra $D(V)^{G'}$ of invariant differential operators under the semi-simple part $G'$ of the reductive group $G$. More precisely we show that $D(V)^{G'}$ is the quotient of a Smith algebra by a completely described two-sided ideal.
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