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Symmetric functions and Hall polynomials. 2nd ed
... , λ l(λ) ], where λ i are positive integers obeying the non-increasing rule λ 1 λ 2 λ 3 . . . λ l(λ) 0. All basic facts about Schur polynomials are listed in [5]. Example of Young diagram is presented in the following picture: = [7, 6, 4, 2, 1, 1] λ = Explicit superintegrable formula for this model reads: ...
... Orthogonal properties of Schur polynomials that are expressed in an elegant way via Cauchy identity [5]: ...
... Operator (24) is a cut-and-join operator [10], however, we call it Hamiltonian of Schur polynomials, due to connection to quantum integrable systems 2 . Formulas (27) for Schur polynomials are called Pierri rules [5]. The sums in the above runs over all possible ways of adding/removing boxes in Young diagram. ...
We develop methods for systematic construction of superintegrable polynomials in matrix/eigenvalue models. Our consideration is based on a tight connection of superintegrable property of Gaussian Hermitian model and algebra in Fock representation. Motivated by this example, we propose a set of assumptions that may allow one to recover superintegrable polynomials. The main two assumptions are box adding/removing rule (Pierri rule) and existence of Hamiltonian for superintegrable polynomials. We detail our method in case of the Gaussian Hermitian model, and then apply it to the cubic Kontsevich model.
... This work started with an observation that the eigenvalues of T on F (d, ℓ) appeared to be non-negative integers. For example, the spectrum of T on F (12, 4) is [1,3,3, 5, 6, 7, 7, 10, 10, 10, 13, 15, 17, 19, 30]. ...
... and may be identified with F via x k = p k k . We refer to [1] for the basic properties of symmetric functions that we review here. ...
... For the Young diagram Λ above, the hook and the leg numbers (d i , q i ), i = 1, . . . , k, are (12, 6), (10, 5), (5, 3), (1,1). ...
We consider two natural gradings on the space of symmetric functions: by degree and by length. We introduce a differential operator T that leaves the components of this double grading invariant and exhibit a basis of bihomogeneous symmetric functions in which this operator is triangular. This allows us to compute the eigenvalues of T, which turn out to be non-negative integers.
... The following theorem yields that the Schur polynomials can be expressed by using the symmetric polynomials or the complete symmetric polynomials. The details can be referred to Macdonald [24]. ...
... Then by (24) and Proposition 4.5, we have ...
... Then by (24) and Proposition 4.5, we have Thus, by Proposition 4.5, X is a T + c -design on G C m,n . ...
The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.
... Similarly, the polynomials Π (λ ′ ,µ ′ )(λ,µ) are the matrix coefficients of the transition matrix from a certain mirabolic Hall-Littlewood basis of Λ ⊗ Λ (introduced in [12]) to the Schur basis, see 4.2. Recall that Λ is isomorphic to the Hall algebra [8] whose natural basis goes to the basis of Hall-Littlewood polynomials. Similarly, Λ ⊗ Λ is naturally isomorphic to a certain mirabolic Hall bimodule over the Hall algebra, and then the natural basis of this bimodule goes to the mirabolic Hall-Littlewood basis, see section 4. The structure constants of this basis, together with Green's formula for the characters of GL N (F q ), enter the computation of the Frobenius traces of the previous paragraph. ...
... So the cardinality of O ρ is a sum of powers of q given by the well known formula for the dimension of the Schubert cells (see e.g. Appendix to Chapter II of [8]). We will denote this cardinality by P ρ . ...
... Once again k = F q . We will freely use the notation of Chapter IV of [8]. In particular, Φ is the set of Frobenius orbits in F q × , or equivalently, the set of irreducible monic polynomials in F q [t] with the exception of f = t. ...
We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji.
... In the Appendix, P. Baumann and the second author link the above construction with the Specht construction and symmetric functions (see [14]). ...
... We write λ n to say that λ is a bipartition of n, and the set of bipartitions of n is denoted by Bip(n). It is well-known that the conjugacy classes of W n are in bijection with Bip(n) (see [9,14]). We defineλ as the signed composition of n obtained by concatenation of λ + and −λ − . ...
... It turns out that ch −1 is an isomorphism of Hopf algebra, whose inverse will be denoted by ch. Then, using arguments of orthogonality and integrality, it can be shown [14,I,Appendix B,9] that the image under ch of the irreducible characters of G n are the elements s λ , where λ ∈ P G is such that |λ| = n. ...
We construct a subalgebra of dimension of the group algebra of a Weyl group of type containing its Solomon descent's algebra but also the Solomon's descent algebra of the symmetric group. This lead us to a construction of the irreducible characters of the hyperoctahedral groups by using a generalized plactic equivalence.
... It has been known [11] that the characters of the irreducible q(n)-modules under consideration in this letter are essentially the Schur Q-functions Q λ (cf. [8]). We remark that the difficult question of finding the character of a general finite-dimensional irreducible q(n)-module has been solved recently by Penkov and Serganova [10]. ...
... The (q(m), q(n))-duality can now be interpreted as a representation theoretic realization of the following well known identity for Schur Q-functions (cf. Macdonald [8]): ...
... The following remarkable theorem is due to Sergeev [11], which will be referred to as (Schur-)Sergeev duality throughout the Letter. We refer the reader to [8] for definitions and properties of the Schur Q-functions Q λ . ...
We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur-Sergeev duality between q(n) and a central extension \Hy_k of the hyperoctahedral group H_k. We show that the zero-weight space of a q(n)-module with highest weight given by a strict partition of n is an irreducible module over the finite group \Hy_n parameterized by . We also discuss some consequences of this Howe duality.
... ]. A more precise construction of this ring can be found in [19] section I.2 (and roughly, the notation of this reference will be followed). ...
... The Macdonald integral basis [19] for the symmetric functions are defined by the following two conditions a) J λ = s∈λ (1 − q a λ (s) t l λ (s)+1 )s λ + µ<λ s µ c µλ (q, t) (s ∈ λ means run over all cells s in λ) b) J λ , J µ qt = 0 f or λ = µ where , qt denotes the scalar product of symmetric functions defined on the power symmetric functions by p λ , p µ qt = δ λµ z λ k 1−q λ k 1−t λ k (z λ is the size of the stablizer of the permutations of cycle structure λ and δ xy = 1 if x = y and 0 otherwise). The coefficients c µλ (q, t) are determined by these two conditions and are rational functions in q and t. ...
... Define the basis H µ [X; t] = H µ [X; 0, t] as Hall-Littlewood symmetric functions. The H µ [X; t] are a transformed version of the Hall-Littlewood polynomials defined in [19] and are analogous to the H µ [X; q, t]. ...
We present two symmetric function operators and that have the property and . These operators are generalizations of the analogous operator and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, and , on standard tableaux such that the q,t Kostka polynomials are given by the sum over standard tableaux of shape \la, K_{\la\mu}(q,t) = \sum_T t^{a_{\mu}(T)} q^{b_{\mu}(T)} for the case when when is two columns or of the form or . This provides proof of the positivity of the (q,t)-Kostka coefficients in the previously unknown cases of K_{\la (32^a1^b)}(q,t) and K_{\la (42^a1^b)}(q,t). The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when is two columns.
... The sets of solutions for Problems P3 and P4 are also the same, namely they are integral points in the above polyhedral cone. This result is due to P. Hall, J. Green and T. Klein, see [Kle1] and [Kle2]; see also [Mac,pg. 100] for the history of this problem. ...
... is true for all split reductive groups over Q. This was the harder of the two implications for GL(m), proved by T. Klein in [Kle1] and [Kle2], see also [Mac,. ...
... was first proved for GL(m) by P. Hall but not published. In fact it follows from a beautiful and elementary observation of J. Green, which is set forth and proved in [Mac,, and which we will outline in §8.6. The description of the solutions to Problem P4 as the set of integral points in the polyhedral cone Sol(P1) (known as the Saturation Conjecture) is due to A. Knutson and T. Tao, [KT] (see also [DW] for a proof using quivers and [KM2] for a proof in the spirit of the present paper using Littelmann's path model). ...
In this paper we apply our results on the geometry of polygons in Cartan subspaces, symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the computation of the structure constants of the Hecke and representation rings associated with a split reductive algebraic group over Q and its complex Langlands' dual. We give a new proof of the ``Saturation Conjecture'' for GL(m) as a consequence of our solution of the corresponding ``saturation problem'' for the Hecke structure constants for all split reductive algebraic groups over Q.
... , ℓ − 1, satisfying the condition λ k+1 ≤ n. For definitions regarding partitions, see [13]. The condition λ k+1 ≤ n is known as the hook condition: in terms of Young diagrams, it means that the diagram of λ should be inside the (k, n)-hook [12]. ...
... with s λ a supersymmetric Schur function [12,13,15,16]. ...
... where ℓ = λ 1 stands for the length of λ ′ . Following the terminology of [13], λ − σ is a vertical strip. ...
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(n|n). The related Gel'fand-Zetlin patterns are based upon the decomposition according to a particular chain of subalgebras of gl(n|n). This chain contains only genuine Lie superalgebras of type gl(k|l) with k and l nonzero (apart from the final element of the chain which is gl(1|0)=gl(1)). Explicit expressions for a set of generators of the algebra on this Gel'fand-Zetlin basis are determined. The results are extended to an explicit construction of a class of irreducible highest weight modules of the general linear Lie superalgebra gl(\infty|\infty).
... Information of our conventions regarding integer partitions (and functional identities which shall be important in the following) is compiled in appendix A, and the refined topological vertex C µνρ (q, t) is defined in (B.1). Using the expression for the latter, we can write (III.15) as where s µ/ν are (skew) Schur functions (see [36] for the definition). The partitions in (III.16) are cyclically identified (e.g. ...
... In performing the calculations in Section III and Section IV, we utilized a number of computational identities. Firstly, we found the following two identities helpful for performing sums of skew Schur functions (see [36] (page 93)) η s η t /µ (x)s η/ν (y) = ∞ i,j=1 ...
... Secondly, we also used the following identities between products over integer partitions and infinite products [36] ∞ i,j=1 ...
We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds at a generic point in the K\"ahler moduli space. These manifolds engineer little string theories in five dimensions or lower and are dual to stacks of M5-branes probing a transverse orbifold singularity. Using the refined topological vertex formalism, we explicitly calculate a generic building block which allows to compute the topological string partition function of as a series expansion in different K\"ahler parameters. Using this result we give further explicit proof for a duality found previously in the literature, which relates for and .
... In this subsection we review some standard facts about symmetric functions. For further details and proofs of the statements reviewed below see [21]. As symmetric functions are labelled by partitions we will begin by reviewing basic features of the latter. ...
... For each choice of λ, µ ∈ P, the matrix elements K λ,µ (q) are polynomials in the parameter q. They are known as Kostka polynomials (see eg Chapter III.6 of [21]) and they will play a central role in our evaluation of the partition function. An explicit combinatoric formula for the Kostka polynomials due to Kirillov and Reshetikhin is given in Appendix B. ...
... The last two listed properties follow easily from the definition (3.15) and the third follows from the invertibility of the change of basis proven in[21] (see Eqn(2.6) in Chapter III of this reference). The first two properties are highly non-trivial and were first proven in[22]. ...
We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large N limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.
... , 1). By using the property of α-determinants developed in Section 2 for α = − 1 k , we show that the wreath determinant is a relative invariant for the action of the wreath product of symmetric groups S k ≀ S n (see [7]) in Section 4. Furthermore, in Section 5, we give an expression of the wreath determinant of kn × n-matrix A by a linear sum of the n-th minor determinants of A labeled by the corresponding rectangular shaped tableaux. In the derivation of this expression, (GL m , GL n )-duality in the sense of [3] provides a guiding principle. ...
... We also give one remark on the background which explains how to get this expression and to understand a structure of the cyclic module U(gl n ) · det (α) (X) ℓ for a general positive integer ℓ in the framework of (GL m , GL n )-duality. Note that the latter closely relates a problem for calculating a certain plethysm [7,4]. ...
... We also put f λ = |STab(λ)| = K λ, (1,...,1) , and denote by ℓ(λ) the depth of the diagram λ. See [1,7] for detailed information on partitions and tableaux. ...
From the irreducible decompositions' point of view, the structure of the cyclic -module generated by the -determinant degenerates when . In this paper, we show that -determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using -duality in the sense of Howe, we obtain an expression of a wreath determinant by a certain linear combination of the corresponding ordinary minor determinants labeled by suitable rectangular shape tableaux. Also we study a wreath determinant analogue of the Vandermonde determinant, and then, investigate symmetric functions such as Schur functions in the framework of wreath determinants. Moreover, we examine coefficients which we call (n,k)-sign appeared at the linear expression of the wreath determinant in relation with a zonal spherical function of a Young subgroup of the symmetric group .
... and λ ′ is the partition conjugate to λ (for partitions we follow the standard notations of [30]). ...
... , m2 ,s , m1 ,s ) and (m 1,s , m 2,s , . . .) are partitions[30], since they consist of non-decreasing non-negative integers. We shall refer to these partitions as the left and right part of row s. ...
The algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is known to be the Lie superalgebra osp(2m+1|2n). The Fock spaces of such systems are then certain lowest weight representations of osp(2m+1|2n). In the current paper, we investigate what happens when the number of parafermions and parabosons becomes infinite. In order to analyze the algebraic structure, and the Fock spaces, we first need to develop a new matrix form for the Lie superalgebra B(n,n)=osp(2n+1|2n), and construct a new Gelfand-Zetlin basis of the Fock spaces in the finite rank case. The new structures are appropriate for the situation . The algebra generated by the infinite number of creation and annihilation operators is , a well defined infinite rank version of the orthosymplectic Lie superalgebra. The Fock spaces are lowest weight representations of , with a basis consisting of particular row-stable Gelfand-Zetlin patterns.
... Let ω be the standard involution of symmetric functions that interchanges elementary symmetric functions and complete symmetric functions (e.g. [Mac,(2.7)]). Note that ω sends s µ to s µ ′ and recall (3.5). ...
... By [CK,Lemma 7.1] or [Mac,(1.7)], for λ = (λ 1 , λ 2 , . . ...
We provide a systematic approach to obtain formulas for characters and Kostant -homology groups of the oscillator modules of the finite dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned -homology computation are worked out.
... For our purposes, a dominance order on partitions is defined as follows: if λ and µ are partitions, then λ ☎ µ if either (1) |µ| > |λ| or (2) |µ| = |λ| and k i=1 λ i ≥ k i=1 µ i for all k > 0. We will write λ ✄ µ to mean that λ ☎ µ and λ = µ. Although the definition of the dominance order on partitions employed here differs from the conventional definition [7] of the dominance order on partitions, when restricted to the partitions of the odd integers {1, 3, . . . , n} or to partitions of the even integers {0, 2, . . . ...
... To each λ-tableau t, associate a unique permutation d(t) ∈ S n−2f by the condition t = t λ d(t). If we refer to the tableau t in (2.1) above for instance, then d(t) = (6, 8) (7,10,9) ...
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra by lifting bases for cell modules of is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B--M--W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori--Hecke algebra of the symmetric group.
... The XX0 model has diverse relations with recent topics of research in mathematical physics such as combinatorial and probabilistic models such as alternating sign matrices [35], random tilings, theory of random walks in lattice and random matrix theory [22], plane partitions and theory of symmetric functions [36], [37], and also topological string theory [38]. ...
... where θ i 's are eigenvalues of the random matrix, S λ is the symmetric Schur function of strict partition λ = (N ≥ λ 1 > λ 2 > ... > λ N f ≥ 0) with λ i = j i − N f + i and λ ′ i = l i − N f + i (i = 0, 1, ..., N f − 1). The matrix integral (50) contains symmetric Schur functions [37], and these functions in the matrix integral play the role of spin operators in XX0 model and in fact they represent and fix the locations of the flipped spins, in the correlation function. We can choose the locations of spin operators as in the partition function of XX0 model, Eq. 3, then we observe that λ i = λ ′ i = 0 and thus the Schur functions in expression (50) become one. ...
We study finite size and temperature XX0 Heisenberg spin chain in weak and strong coupling regimes. By using an elegant connection of the model to integrable combinatorics and probability, we explore and interpret a possible phase structure of the model in asymptotic limit: the limit of large inverse temperature and size. First, partition function and free energy of the model are derived by using techniques and results from random matrix models and nonintersecting Brownian motion. We show that, in the asymptotic limit, partition function of the model, written in terms of matrix integral, is governed by the Tracy-Widom distribution. Second, the exact analytic results for the free energy, which is obtained by the asymptotic analysis of the Tracy-Widom distribution, indicate a completely new and sophisticated phase structure of the model. This phase structure consists of second- and third-order phase transitions. Finally, to shed light on our new results, we provide a possible interpretation of the phase structure in terms of dynamical behavior of magnons in the spin chain. We demonstrate distinct features of the phases with schematic spin configurations which have definite features in each region of the phase diagram.
... Note that in the theory of symmetric functions [13,14] an expression like (11) relates the complete homogeneous symmetric functions to the so called power sum symmetric functions or Schur polynomials, so we can use the combinatorics of that theory to write down the explicit (invertible) relation between the functionals ψ k [J] and the S i [J], as homogeneous polynomials of order k in the latter. This is given by ...
... It is not difficult to verify that the non-linear relation between the ψ k 's and S i 's resulting from (62) is given by (13), and that the ψ k 's acquire a primitive coproduct ...
The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones.
... Our notations will be those of [11] and [7]. If S is a set of words, we denote by S = w∈S w its characteristic series. ...
... where the complete homogeneous functions h n (X) are defined by σ t (X) = n≥1 (1−tx n ) −1 (cf. [11]). The q n 's are connected to the power-sums p n (X) = i x n i by p n = d|n dq n/d d (2) and condition (W2) can be regarded as expressing the familiar properties of power sums ...
The construction of the universal ring of Witt vectors is related to Lazard's factorizations of free monoids by means of a noncommutative analogue. This is done by associating to a code a specialization of noncommutative symmetric functions.
... We first check the formula for columns 1 n , by using the recursive formula for the antisymmetrizers y 1 n . We note [11,Ch.1] that the right hand side in Theorem 2.3 is the Schur polynomial in the y 1 n h . We then proceed recursively on the number of cells. ...
... The recursive formula (11) can be deduced. It can be written ...
We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas.
... and is contained in Sym k (C d ) ⊗ Sym n−k (C d ). By Pieri's formula (see [5,Theorem 40.4] or [38,Equation 5.16]), we have ...
... In the setup of Theorem 19, suppose µ L 2 and µ L 1 ⊗L 2 are such that they define approximate coherent state POVMs with incompleteness bounded by δ. That is, 38 Then the statement of Theorem 19 holds for a δ-approximately normalized mixture distribution with error ǫ ≤ 2R + 4δ where ...
The sum-of-squares hierarchy of semidefinite programs has become a common tool for algorithm design in theoretical computer science, including problems in quantum information. In this work we study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem, and geometric quantization of compact K\"ahler manifolds (such as complex projective space , the set of all pure states in a -dimensional Hilbert space). We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function to obtain a finite-dimensional matrix, finding its top eigenvector, and then (possibly nonconstructively) rounding it by using a version of the Husimi quasiprobability distribution. Dually, we recover most known quantum de Finetti theorems by doing the same steps in the reverse order: a quantum state is first approximated by its Husimi distribution, and then quantized to obtain a separable state approximating the original one. In cases when there is a transitive group action on the manifold we give some new proofs of existing de Finetti theorems, as well as some applications including a new version of Renner's exponential de Finetti theorem proven using the Borel--Weil--Bott theorem, and hardness of approximation results and optimal degree-2 integrality gaps for the basic SDP relaxation of \textsc{Quantum Max-d-Cut} (for arbitrary d). We also describe how versions of these results can be proven when there is no transitive group action. In these cases we can deduce some error bounds for the HSoS hierarchy on complex projective varieties which are smooth.
... A partition λ of a positive integer n is a finite non-increasing sequence of positive integers (λ 1 , λ 2 , . . . , λ m ) satisfying λ 1 + λ 2 + ⋅ ⋅ ⋅ + λ m = n (see [18,25]). We denote it by λ ⊢ n. ...
Simultaneous core partitions have been widely studied in the past 20 years. In 2013, Amdeberhan gave several conjectures on the number, the average size, and the largest size of (t,t+1)-core partitions with distinct parts, which was proved and generalized by Straub, Xiong, Nath-Sellers, Zaleski-Zeilberger, Paramonov, and many other mathematicians. In this paper, we introduce a proper self-conjugate partition analog of (t,t+1)-core partitions with distinct parts, and derive the number, the average size, and the largest size for such core partitions.
... The extension of Macdonald polynomials [1] to the super-case [2,3] and [4,5] appears to be applicable to studies of super-Yangians like Y ( gl 1,1 ) [6][7][8] and their super-DIM generalizations [5,9,10]. Surely, a modification (introduction of Grassmann variables) of the original Macdonald construction sheds some light on peculiarities of the original construction itself: we might try to question whether an appearance of algebraic and combinatorial structures accompanying Macdonald polynomials is accidental and may be naturally extended to other similar constructions. ...
The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables new Grassmann time variables are introduced, and the Hamiltonian is represented as a differential operator acting on a space of functions of both types of variables and . Eigenfunctions for this Hamiltonian are a suitable generalization of Macdonald polynomials to super-Macdonald polynomials discussed earlier in the literature. Peculiarities of the construction in comparison to the canonical bosonic case are discussed.
... (3) Schur polynomials have deep connections with combinatorics, geometry and representation theory ( [Mac95,§I], [Fu97], [Fu98,§14], [St99,§7] , which is a key fact in the proof in [FL83]. ...
Combinatorial ideas are developed in this article to study Chern numbers on ample and numerically effective vector bundles. An effective lower bound for Chern numbers of ample vector bundles is established, which makes some progress towards a long-standing question. Along this line we prove that Chern numbers on nef vector bundles obey reverse dominance ordering, which improves upon some classical and recent results. We propose a simultaneous positivity question on (signed) Chern numbers of compact complex or K\"{a}hler manifolds whose (co)tangent bundles are semipositive in various senses, and show that it holds true for compact homogeneous complex manifolds.
... We also scratch the surface of the link between Schur functions and the representation theory of the symmetric group Sym(n). We refer the reader to [40] for more information about Schur functors, and to [22,30] for more information about representations of Sym(n). Finally, plethysms also occur in other areas, such as τ -rings [19] and physics [42]. ...
We establish a strong link between two open problems: determining the Reidemeister spectrum of free nilpotent groups and determining the coefficients in the Schur expansion of plethysms of Schur functions. Specifically, we show that the expressions occurring in the computations for the Reidemeister spectrum are sums of plethysms of the form , where g is a Schur function or a Schur positive function.
... Vandermonde polynomials are central to the theory of alternating polynomials. In fact, any alternating polynomial is divisible by the Vandermonde polynomial [11,6]. Further, Vandermonde matrix and Vandermonde polynomial occur very often in the theory of error correcting codes and are useful in Lagrangian interpolation. ...
An n-variate Vandermonde polynomial is the determinant of the n x n matrix where the ith column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and occur in Lagrangian polynomial interpolation as well as in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. Firstly, we consider the problem of testing if a given polynomial is linearly equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorithm to test if the polynomial f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when the polynomial f is given as a black-box our algorithm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and show that the associated Lie algebra is simple.
... The boson-fermion correspondence. Denote Λ the algebra of symmetric functions, that is, the projective limit of the Q(q)-algebras of symmetric polynomials in finitely many indeterminates [21,Chapter 1]: ...
We explain how the action of the Heisenberg algebra on the space of q-deformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the boson-fermion correspondence and looking at the action of the Schur functions at q = 0. In addition, we give the explicit formula for computing this crystal in full generality.
... , λ ℓ ). The numbers λ i (1 ≤ i ≤ ℓ) are called the parts and 1≤i≤ℓ λ i the size of the partition λ (see [12,18]). Each partition λ is identified with its Young diagram, which is an array of boxes arranged in leftjustified rows with λ i boxes in the i-th row. ...
Amdeberhan's conjectures on the enumeration, the average size, and the largest size of (n,n+1)-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of and -core partitions with distinct parts, respectively. Let be the size of a uniform random (s,t)-core partition with distinct parts when s and t are coprime to each other. Some explicit formulas for the k-th moments and were given by Zaleski and Zeilberger when k is small. Zaleski also studied the expectation and higher moments of and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the k-th moments of and in this paper, by studying the beta sets of core partitions. In particular, we show that these k-th moments are asymptotically some polynomials of n with degrees at most 2k, when d is given and n tends to infinity. Moreover, when d=1, we derive that the k-th moment of is asymptotically equal to when n tends to infinity. The explicit formulas for the expectations and are also given. The -core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of .
... MacDonald [31] is an excellent reference for the following discussion. Recall skew-Schur polynomials are defined as: for any µ ⊂ λ, ...
This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee interaction beyond the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.
... Now let us consider that case k = 2. It is known (see [Mac,I.8]) that ...
We investigate the representation of a symmetric group on the homology of its Quillen complex at a prime p. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of p-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of on the top homology group of the corresponding hypergraph matching complex when . Our conjecture follows from work of Bouc when p=2, and we prove the conjecture when p=3.
... where the two last sums are over all partitions and for the equality in the middle we have used Cauchy-Littlewood formula [50,Chapter 1]. An element r ∈ B acts on the set of hypergeometric tau-functions by action on the first factor via r(Ĵ) ⊗ Id. ...
We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y. We propose an argument to prove this statement conditionally to a mild version of symplectic invariance for the 1-hermitian matrix model, which is believed to be true but has not been proved yet. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters are strictly monotone double Hurwitz numbers with ramifications above and above 0. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.
... where the positive integers K λµ are the Kostka numbers (see [22]), satisfying K λλ = 1. We also define, letting λ be a partition with at most r parts with highest part λ 1 < m, which we extend to have r parts by adding 0 where necessary: ...
We use the theory of Gr\"obner-Shirshov bases for ideals to construct linear bases for graded local Weyl modules for the (hyper) current and the truncated current algebras associated to the finite-dimensional complex simple Lie algebra . The main result is a characteristic-free construction of bases for this important family of modules for the hyper current -algebra. In the positive characteristic setting this work represents the first construction in the literature. In the characteristic zero setting, the method provides a different construction of the Chari-Pressley (also Kus-Littelmann) bases and the Chari-Venkatesh bases for local Weyl modules for the current -algebra. Our construction allows us to obtain new bases for the local Weyl modules for truncated current -algebras with very particular properties.
... One has moreover (see e.g. [18]) s λ = a λ+δ /a δ , hence p µ a δ = λ n χ λ µ a λ+δ and in particular, ...
We define the L-measure on the set of Dirichlet characters as an analogue of the Plancherel measure, once considered as a measure on the irreducible characters of the symmetric group. We compare the two measures and study the limit in distribution of characters evaluations when the size of the underlying group grows. These evaluations are proven to converge in law to imaginary exponentials of a Cauchy distribution in the same way as the rescaled windings of the complex Brownian motion. This contrasts with the case of the symmetric group where the renormalised characters converge in law to Gaussians after rescaling (Kerov Central Limit Theorem).
... Define the power sum symmetric function p n (x) = i x n i . As in [12], we can uniquely express the p n (x) in terms of the e n (x) and visa-versa. Thus, the power sum symmetric functions constitute yet another Noetherian transcendence basis for Λ and Artinian transcendence basis for Λ * . ...
We define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized. As an application, we compute the algebra of symmetric functions with nonnegative real exponents. The applications to logarithmic series and the Umbral calculus are deferred to another paper. On d\'efinit ici les alg\`ebres Artinienne et Noetherienne comme \'etant des alg\`ebres constitu\'ees des s\'eries formelles \`a exposants pas n\'ecessairement entiers. On definit sur ces alg\`ebres toutes les op\'erations classiques et on les caracterise. Comme exemple d'exploitation de cette th\'eorie, on s'interesse \`a alg\`ebre de fonctions sym\'etriques à exponsants r\`eels en nonn\'egatifs. Une autre publication est consacr\'ee aux applications aux series logarithmiques et au calcul ombral.
... This paper has been motivated by an attempt to extend the classical representation theory of the symmetric group [17,18,21,25,27] to arbitrary Coxeter groups. Our method of generalization is through the explicit representation matrices, following ideas from [5,22,29]. ...
An elementary approach to the construction of Coxeter group representations is presented.
... A form of bispectrality may also be seen in terms of special families of multivariable orthogonal polynomials (rather than a single function Ψ depending on spectral parameters). In the case of the root system A l , these are the Jack polynomials, and for other root systems they are the multivariable Jacobi polynomials, which admit Pieri-type formulas that can be interpreted as bispectrality between the CMS Hamiltonians and difference operators acting on the weights indexing the polynomials [2,15,20]. ...
We show that a Sergeev–Veselov difference operator of rational Macdonald–Ruijsenaars (MR) type for the deformed root system BC(l,1) preserves a ring of quasi-invariants in the case of non-negative integer values of the multiplicity parameters. We prove that in this case the operator admits a (multidimensional) Baker–Akhiezer eigenfunction, which depends on spectral parameters and which is, moreover, as a function of the spectral variables an eigenfunction for the (trigonometric) generalised Calogero–Moser–Sutherland (CMS) Hamiltonian for BC(l,1). By an analytic continuation argument, we generalise this eigenfunction also to the case of more general complex values of the multiplicities. This leads to a bispectral duality statement for the corresponding MR and CMS systems of type BC(l,1).
Let O be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite O-module M of rank two. The main emphasis is on the interaction between the different groups and their representations. An induction scheme is developed in order to study the whole family of these groups coherently. The results obtained depend on the ring O in a very weak manner, mainly through the degree of the residue field. In particular, a uniform description of the irreducible representations of GL(2,O/P^k) is obtained, where P is the maximal ideal of O.
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.
We study the dynamical analogue of the matrix algebra M(n), constructed from a dynamical R-matrix given by Etingof and Varchenko. A left and a right corepresentation of this algebra, which can be seen as analogues of the exterior algebra representation, are defined and this defines dynamical quantum minor determinants as the matrix elements of these corepresentations. These elements are studied in more detail, especially the action of the comultiplication and Laplace expansions. Using the Laplace expansions we can prove that the dynamical quantum determinant is almost central, and adjoining an inverse the antipode can be defined. This results in the dynamical GL(n) quantum group associated to the dynamical R-matrix. We study a *-structure leading to the dynamical U(n) quantum group, and we obtain results for the canonical pairing arising from the R-matrix.
We compute the cycle index sum of the symmetric group action on the homology of the configuration spaces of points in a Euclidean space with the condition that no k of them are equal.
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.
New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).
We show the classical q-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1+q. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. Its rank generating function is the q-Stirling number . The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. Letting we give a bijective argument showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.
We study an extension of the Seiberg-Witten theory of 5d supersymmetric Yang-Mills on . We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted 5d supersymmetric Yang-Mills in the background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential.
Let denote the symmetric group on n elements, and a symmetric subset of permutations. Aldous' spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if is a set of transpositions, then the second eigenvalue of the Cayley graph is identical to the second eigenvalue of the Schreier graph on n vertices depicting the action of on . Inspired by this seminal result, we study similar questions for other types of sets in . Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [2008], we show that for large enough n, if is a full conjugacy class, then the second eigenvalue of is roughly identical to the second eigenvalue of the Schreier graph depicting the action of on ordered 4-tuples of elements from . We further show that this type of result does not hold when is an arbitrary normal set, but a slightly weaker one does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set , which yields surprisingly strong consequences.
The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related operator of Bergeron and Garsia (1999) to basic symmetric functions. The first discovery of this type was the (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005), which relates the expression to parking functions. In (2007), Loehr and Warrington conjectured a similar expression for in terms of labeled square paths. In this paper, we extend Haglund and Loehr's (2005) notion of schedules to labeled square paths and apply this extension to prove the Square Paths Conjecture.
Making use of a Howe duality involving the infinite-dimensional Lie superalgebra \hgltwo and the finite-dimensional group we derive a character formula for a certain class of irreducible quasi-finite representations of \hgltwo in terms of hook Schur functions. We use the reduction procedure of \hgltwo to to derive a character formula for a certain class of level 1 highest weight irreducible representations of , the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra . These modules turn out to form the complete set of integrable -modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible -modules may be written as a sum of products of hook Schur functions.
A series of bilinear identities on the Schur symmetric functions is obtained with the use of Pluecker relations.
Let denote the alternating and the symmetric groups on 1,...,n. MacMahaon's theorem, about the equi-distribution of the length and the major indices in , has received far reaching refinements and generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and followers. Our main goal is to find analogous statistics and identities for the alternating group . A new statistic for , {\it the delent number}, is introduced. This new statistic is involved with new equi-distribution identities, refining some of the results of Foata-Schutzenberger and Garsia-Gessel. By a certain covering map , such identities are `lifted' to , yielding the corresponding equi-distribution identities.
We consider the (direct sum over all n ∈ N of the) K-theory of the seminilpotent commuting variety of gln, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e., a particular Z[q±1 1 , q±1 2 ]-submodule of the equivariant K-theory of a point) and the second as the Z[q±1 1 , q±1 2 ]-algebra generated by certain elements { ¯Hn,d}(n,d)∈N×Z. As the shuffle algebra over Q(q1, q2) has long been known to be isomorphic to half of an algebra known as quantum toroidal gl1, we thus obtain a description of an important integral form of the quantum toroidal algebra.
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