This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory $\mathfrak{gl}_n \C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern.
As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.
We give a survey on the Littlewood-Richardson rule. Using Gelfand-Tsetlin patterns as the main machinery of our analysis, we study the interrelationship of various combinatorial descriptions of the Littlewood-Richardson rule.
We give a crystal structure on the set of Gelfand-Tsetlin patterns which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form, expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between Gelfand-Tsetlin patterns and semistandard Young tableaux is a crystal isomorphism.
This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity. Some properties of the Gelfand-Tsetlin basis are derived via an algorithm for solving certain equations on the lattice of…
We exploit the idea that the character of an irreducible finite dimensional $\mathfrak{gl}_n$-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits of Gelfand-Tsetlin polytopes. Namely, we show that under the relevant substitution the integer point transforms of all but $n!$ vertices vanish, the remaining ones being the summands in Weyl's formula.
In their study of the equivariant K-theory of the generalized flag varieties $G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called the alcove paths model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove paths model and the Gelfand-Tsetlin patterns model for type $A$.
Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin polytopes (1950) and the Feigin-Fourier-Littelmann polytopes (2010), which arise in the representation theory of the special linear Lie algebra. We then…
We determine the diameter of the 1-skeleton and the combinatorial automorphism group of any Gelfand-Tsetlin polytope GT$_λ$ associated to an integer partition $λ.$
Macdonald polynomials are an important class of symmetric functions, with connections to many different fields. Etingof and Kirillov showed an intimate connection between these functions and representation theory: they proved that Macdonald polynomials arise as (suitably normalized) vector-valued characters of irreducible representations of quantum groups. In this paper, we provide a branching rule for these characters. The coefficients are expressed in terms of skew Macdonald polynomials with plethystic substitutions. We use our branching rule to give an expansion of the characters with respe…
Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope $\mathrm{GT}(λ)$ is equal to the dimension of the corresponding irreducible representation of $GL(n)$. It is well-known that the Gelfand-Tsetlin polytope is a marked order polytope; the authors have recently shown it to be a flow polytope. In this paper, we draw corollaries from this result and establish a general theory connecting marked order polytopes and flow polytopes.
We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for each finite-dimensional representation of $\mathfrak{sl}(n,\mathbb{C})$ associated to a given skew Schur function. Our constructions use diamond-colored distributive lattices of skew-shaped semistandard tableaux that generalize some classical Gelfand--Tsetlin (GT) lattices. Our constructions take place within the context of a certain programmatic study of p…
We shall present a ``linear algebraic'' proof (involving some calculations in the algebra of linear operators on a vector space of polynomials and some manipulations of determinants) of the formula for the enumeration of symmetric Gelfand--Tsetlin patterns with fixed bottom row, which was proved by Tri Lai in the context of enumerating symmetric lozenge tilings of a ``halved'' hexagon with ``dents''.
The Gelfand-Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand-Tsetlin graph has at least three incarnations --- as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinite-dimensional unitary group, and as the set of doubly infinite totally positive sequences. An old deep result due to Albert Edrei and Dan Voiculescu provides an explicit description of the boundary; it can be realized as a region in an infinite-dimensional coordinate space. The paper contains …
Relation Gelfand-Tsetlin $\mathfrak{gl}_n$-modules were introduced in [FRZ19], and are determined by some special directed graphs and Gelfand-Tsetlin characters. In this work we constructed polyhedra associated with the class of relation modules, which includes as a particular case, any classical Gelfand-Tsetlin polytope. Following the ideas presented in [LM04] we give a characterization of $d$-faces of the associated polyhedra in terms of a matrix related to the corresponding graph.
Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand-Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or ``hives'') on a ``convex part'' of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applicat…
Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of $\mathfrak{gl}_n(\mathbb{C})$. The integer point transform of the Gelfand-Tsetlin polytope $\mathrm{GT}(λ)$ projects to the Schur function $s_λ$. Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials $\mathfrak{S}_{w}$ corresponding to Grassmannian permutations. For any permutation $w \in S_n$ with column-convex Rothe diagram, we construct a polytope $\mathcal{P}_{w}$ whose integer point transform project…
A probabilistic bijection between twenty-vertex configurations with a free west boundary and Gelfand-Tsetlin patterns avoiding three equal entries in a row
Let $P$ be the Gelfand--Tsetlin polytope defined by the skew shape $λ/μ$ and weight $w$. In the case corresponding to a standard Young tableau, we completely characterize for which shapes $λ/μ$ the polytope $P$ is integral. Furthermore, we show that $P$ is a compressed polytope whenever it is integral and corresponds to a standard Young tableau. We conjecture that a similar property hold for arbitrary $w$, namely that $P$ has the integer decomposition property whenever it is integral. Finally, a natural partial ordering on GT-polytopes is introduced that provides information about integrality …
A continuation of the joint work by Vadim Gorin and the author, J. Funct. Anal. 270 (2016), 375-418; arXiv:1504.06832. The extended Gelfand-Tsetlin graph, introduced in that paper, is a novel combinatorial object. Its q-boundary is formed by infinite point configurations on a two-sided q-lattice. The q-boundary carries a continuous family of probability measures that are a q-analogue of the so-called zw-measures, which originated in the problem of harmonic analysis on the infinite-dimensional unitary group. In the present paper, it is proved that certain transition Markov kernels, linked to th…
In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonneg…
We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052-3132, arXiv:1301.0634], and are expected to provide tools for the study of statistical mechanical models, representation theory and random matrices. As first application of our formulas, we characterize the boundary of the $(q,t)$-deformation of the Gelfand-Tsetlin graph when $t=q^θ$ and $θ$ is a positive integer.
The de Bruijn-Tengbergen-Kruyswijk (BTK) construction is a simple algorithm that produces an explicit symmetric chain decomposition of a product of chains. We linearize the BTK algorithm and show that it produces an explicit symmetric Jordan basis (SJB). In the special case of a Boolean algebra the resulting SJB is orthogonal with respect to the standard inner product and, moreover, we can write down an explicit formula for the ratio of the lengths of the successive vectors in these chains (i.e., the singular values). This yields a new, constructive proof of the explicit block diagonalization …
Using Okounkov's $q$-integral representation of Macdonald polynomials we construct an infinite sequence $Ω_1,Ω_2,Ω_3,\dots$ of countable sets linked by transition probabilities from $Ω_N$ to $Ω_{N-1}$ for each $N=2,3,\dots$. The elements of the sets $Ω_N$ are the vertices of the extended Gelfand-Tsetlin graph, and the transition probabilities depend on the two Macdonald parameters, $q$ and $t$. These data determine a family of Markov chains, and the main result is the description of their entrance boundaries. This work has its origin in asymptotic representation theory. In the subsequent paper…
By rewriting the famous hook-content formula it easily follows that there are $\prod\limits_{1 \le i < j \le n} \frac{k_j - k_i + j -i}{j-i}$ semistandard tableaux of shape $(k_n,k_{n-1},...,k_1)$ with entries in $\{1,2,...,n\}$ or, equivalently, Gelfand-Tsetlin patterns with bottom row $(k_1,...,k_n)$. In this article we introduce certain sequences of labeled trees, the signed enumeration of which is also given by this formula. In these trees, vertices as well as edges are labeled, the crucial condition being that each edge label lies between the vertex labels of the two endpoints of the edge…
The thesis focuses on processes on symplectic Gelfand-Tsetlin patterns. In chapter 4, a process with dynamics inspired by the Berele correspondence [Ber86] is presented. It is proved that the shape of the pattern is a Doob $h$-transform of independent random walks with $h$ given by the symplectic Schur function. This is followed by an extension to a $q$-weighted version. This randomised version has itself a branching structure and is related to a $q$-deformation of the $so_{2n+1}$-Whittaker functions. In chapter 5, we present a fully randomised process. This process $q$-deforms a process propo…
The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that t…
A Gelfand-Tsetlin scheme of depth N is a triangular array with m integers at level m, m=1,...,N, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed N-th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows…
We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_n(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n…
Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain-order polytopes as Newton-Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal …
We discuss the problem of counting vertices in Gelfand-Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas.
One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this paper, we explicitly describe string parametrizations of opposite Demazure crystals, which give a natural generalization of reduced dual Kogan faces. We also relate reduced Kogan faces with Demazure crystals using the theory of …
For Grassmann varieties, we explain how the duality between the Gelfand-Tsetlin polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes arises from different positive structures.
A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes. This solves the open problem (2) posed by Gusev, Kritchenko, and Timorin in [GKT].
The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric group which encodes the character of a certain irreducible representation of a Lie group of type $A$. In this paper, we provide a recursive formula for the number of reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalent to a Gelfand-Cetlin polytope. The recursive formula involves the number of standard Young tableaux…
Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of Abe-Horiguchi-Masuda-Murai-Sato and Abe-DeDieu-Galetto-Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand-Zetlin polytope $\mathrm{GZ}(λ)$ for $λ=(λ_1,λ_2,\ldots,λ_n)$. The main results of this manuscript use and generalize tools developed by Anderson-Tymoczko, Kiritchenko-Smirnov-Timorin, and Postnikov, in orde…
For a reduced word ${\bf i}$ of the longest element in the Weyl group of $\mathrm{SL}_{n+1}(\mathbb{C})$, one can associate the string cone $C_{\bf i}$ which parametrizes the dual canonical bases. In this paper, we classify all ${\bf i}$'s such that $C_{\bf i}$ is simplicial. We also prove that for any regular dominant weight $λ$ of $\mathfrak{sl}_{n+1}(\mathbb{C})$, the corresponding string polytope $Δ_{\bf i}(λ)$ is unimodularly equivalent to the Gelfand-Cetlin polytope associated to $λ$ if and only if $C_{\bf i}$ is simplicial. Thus we completely characterize Gelfand-Cetlin type string poly…
We prove that the boundary of the Hall-Littlewood $t$-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when $1/t$ is a prime $p$ we use this to recover results of Bufetov-Qiu and Assiotis on infinite $p$-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions. Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain …
A combinatorial study of multiple $q$-integrals is developed. This includes a $q$-volume of a convex polytope, which depends upon the order of $q$-integration. A multiple $q$-integral over an order polytope of a poset is interpreted as a generating function of linear extensions of the poset. Specific modifications of posets are shown to give predictable changes in $q$-integrals over their respective order polytopes. This method is used to combinatorially evaluate some generalized $q$-beta integrals. One such application is a combinatorial interpretation of a $q$-Selberg integral. New generatin…
