Semistandard Young Tableaux Research Articles

Basis for non-factorizable superamplitudes in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetry

In this paper we develop a semi-standard Young tableau (SSYT) approach to construct a basis of non-factorizable superamplitudes in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 massless supersymmetry. This amplitude basis can be directly translated to a basis for higher dimensional supersymmetric operators, yielding both the number of independent operators and their form. We deal with distinguishable (massless) chiral/vector superfields at first, then generalize the result to the indistinguishable case. Finally, we discuss the advantages and disadvantages of this method compared to the previously studied Hilbert series approach.

Free Action and Cyclic Sieving on Skew Semi-standard Young Tableaux

In this note, we provide a short proof of Theorem 4.3 in the paper titled Crystals, semistandard tableaux and cyclic sieving phenomenon, by Y.-T. Oh and E. Park, which concerns a cyclic sieving phenomenon on semi-standard Young tableaux. We also extend their result to skew shapes.

Schubert Polynomials, the Inhomogeneous TASEP, and Evil-Avoiding Permutations

Abstract Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\ldots ,n\}$; the possible arrangements of particles in sites thus correspond to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on $S_n$, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise, nothing happens.) When $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials [18], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [4, 5]. In the case of general $y_i$, Cantini [7] showed that $n$ of the $n!$ states have probabilities proportional to double Schubert polynomials. In this paper, we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213,$ and $3214$. We show that there are $\frac {(2+\sqrt {2})^{n-1}+(2-\sqrt {2})^{n-1}}{2}$ evil-avoiding permutations in $S_n$, and for each evil-avoiding permutation $w$, we give an explicit formula for the steady state probability $\psi _w$ as a product of double Schubert polynomials. (Conjecturally, all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When $y_i=0$ for all $i$, we give multiline queue formulas for the $\textbf {z}$-deformed steady state probabilities and use this to prove the monomial factor conjecture from [18]. Finally, we show that the Schubert polynomials arising in our formulas are flagged Schur functions, and we give a bijection in this case between multiline queues and semistandard Young tableaux.

Alcove Paths and Gelfand–Tsetlin Patterns

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 path 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 path model and the Gelfand–Tsetlin pattern model for type A.

Symplectic Keys and Demazure Atoms in Type C

We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.

Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian

Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studying Gröbner degenerations of their corresponding ideals. These degenerations are parametrised by block diagonal matching fields in the sense of Sturmfels [33]. We associate a weight vector to each block diagonal matching field and study its corresponding initial ideal. In particular, we characterise when such ideals are toric, hence providing a family of toric degenerations for Richardson varieties. Given a Richardson variety $$X_{w}^v$$ and a weight vector $$\mathbf{w}_\ell $$ arising from a matching field, we consider two ideals: an ideal $$G_{k,n,\ell }|_w^v$$ obtained by restricting the initial of the Plücker ideal to a smaller polynomial ring, and a toric ideal defined as the kernel of a monomial map $$\phi _\ell |_w^v$$ . We first characterise the monomial-free ideals of form $$G_{k,n,\ell }|_w^v$$ . Then we construct a family of tableaux in bijection with semi–standard Young tableaux which leads to a monomial basis for the corresponding quotient ring. Finally, we prove that when $$G_{k,n,\ell }|_w^v$$ is monomial-free and the initial ideal in $$_{\mathbf{w}_\ell }(I(X_w^v))$$ is quadratically generated, then all three ideals in $$_{\mathbf{w}_\ell }(I(X_w^v))$$ , $$G_{k,n,\ell }|_w^v$$ and ker $$(\phi _\ell |_w^v)$$ coincide, and provide a toric degeneration of $$X_w^v$$ .

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta functions by an analogue of Weyl group multiple Dirichlet series in the sense of Bump et al. By combining with a result of Nakasuji, Phuksuwan and Yamasaki, our theorems yield a new method of finding functional relations among zeta-functions of root systems.

Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases

We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the metaplectic representation of $\mathfrak{osp}(1|2n)$. We construct a new polynomial basis for $L_n(p)$ arising from the embedding $\mathfrak{osp}(1|2np) \supset \mathfrak{osp}(1|2n)$. The basis vectors of $L_n(p)$ are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra valued polynomials with integer coefficients in $np$ variables. Using combinatorial properties of these tableau vectors it is deduced that they form indeed a basis. The computation of matrix elements of a set of generators of $\mathfrak{osp}(1|2n)$ on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau.

Arctic Curves Phenomena for Bounded Lecture Hall Tableaux

Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded tableaux thanks to two other combinatorial models: non-intersecting paths on a graph whose faces are squares and pentagons and dimer models on a lattice whose faces are hexagons and octagons. We use the tangent method to investigate the arctic curve in the model of non-intersecting lattice paths with fixed starting points and ending points distributed according to some arbitrary piecewise differentiable function. We then study the dimer model and use an ansatz to guess the asymptotics of the inverse of the Kasteleyn, which confirm the arctic curve computed with the tangent method for two examples.

Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

We study Gröbner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial theory to show that block diagonal matching fields give rise to toric degenerations. Our methods and results use the combinatorics of permutations associated to Schubert varieties, matching fields and their corresponding tableaux.

Outliers of random perturbations of Toeplitz matrices with finite symbols

Consider an $$N\times N$$ Toeplitz matrix $$T_N$$ with symbol $${{\varvec{a}} }(\lambda ) := \sum _{\ell =-d_2}^{d_1} a_\ell \lambda ^\ell $$ , perturbed by an additive noise matrix $$N^{-\gamma } E_N$$ , where the entries of $$E_N$$ are centered i.i.d. random variables of unit variance and $$\gamma >1/2$$ . It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as $$N\rightarrow \infty $$ , to the law of $${{\varvec{a}}}(U)$$ , where U is distributed uniformly on $${\mathbb {S}}^1$$ . In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N-independent) distance from $${{\varvec{a}}}({\mathbb {S}}^1)$$ . We prove that there are no outliers outside $$\mathrm{spec} \, T({{\varvec{a}}})$$ , the spectrum of the limiting Toeplitz operator, with probability approaching one, as $$N \rightarrow \infty $$ . In contrast, in $$\mathrm{spec}\, T({{\varvec{a}}}){\setminus } {{\varvec{a}}}({{\mathbb {S}}}^1)$$ the process of outliers converges to the point process described by the zero set of certain random analytic functions. The limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d. having the same law as that of $$E_N$$ . The coefficients in the linear combination depend on the roots of the polynomial $$P_{z, {{\varvec{a}}}}(\lambda ):= ({{\varvec{a}}}(\lambda ) -z)\lambda ^{d_2}$$ and semi-standard Young Tableaux with shapes determined by the number of roots of $$P_{z,{{\varvec{a}}}}(\lambda )=0$$ that are greater than one in moduli.

GENERALIZED ORBITAL VARIETIES FOR MIRKOVIĆ–VYBORNOV SLICES AS AFFINIZATIONS OF MIRKOVIĆ–VILONEN CYCLES

We show that generalized orbital varieties for Mirkovic-Vybornov slices can be indexed by semi-standard Young tableaux. We also check that the Mirkovic-Vybornov isomorphism sends generalized orbital varieties to (dense subsets of) Mirkovic-Vilonen cycles, such that the (combinatorial) Lusztig datum of a generalized orbital variety, which it inherits from its tableau, is equal to the (geometric) Lusztig datum of its MV cycle.

A Crystal on Decreasing Factorizations in the 0-Hecke Monoid

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commu\-tative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.

The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.

Reverse plane partitions of skew staircase shapes and q-Euler numbers

Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two q-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain q-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of the same shape. One conjecture is a determinantal formula for the number of pleasant diagrams in terms of Schröder paths and the other conjecture is a determinantal formula for the generating function for RPPs of a skew staircase shape in terms of q-Euler numbers.In this paper, we show that the results of Morales, Pak and Panova on the q-Euler numbers can be derived from previously known results due to Prodinger by manipulating continued fractions. These q-Euler numbers are naturally expressed as generating functions for alternating permutations with certain statistics involving maj. It has been proved by Huber and Yee that these q-Euler numbers are generating functions for alternating permutations with certain statistics involving inv. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving maj to the statistic involving inv. We also prove the aforementioned two conjectures of Morales, Pak and Panova.

The Diameter and Automorphism Group of Gelfand–Tsetlin Polytopes

Gelfand–Tsetlin polytopes arise in representation theory and algebraic combinatorics. One can construct the Gelfand–Tsetlin polytope $$\mathrm{GT}_\lambda $$ for any partition $$\lambda = (\lambda _1,\ldots ,\lambda _n)$$ of weakly increasing positive integers. The integral points in a Gelfand–Tsetlin polytope are in bijection with semi-standard Young tableau of shape $$\lambda $$ and parametrize a basis of the $$\mathrm{GL}_n$$ -module with highest weight $$\lambda $$ . The combinatorial geometry of Gelfand–Tsetlin polytopes has been of recent interest. Researchers have created new combinatorial models for the integral points and studied the enumeration of the vertices of these polytopes. In this paper, we determine the exact formulas for the diameter of the 1-skeleton, $$\mathrm{diam}(\mathrm{GT}_\lambda )$$ , and the combinatorial automorphism group, $$\mathrm{Aut}(\mathrm{GT}_\lambda )$$ , of any Gelfand–Tsetlin polytope. We exhibit two vertices that are separated by at least $$\mathrm{diam}(\mathrm{GT}_\lambda )$$ edges and provide an algorithm to construct a path of length at most $$\mathrm{diam}(\mathrm{GT}_\lambda )$$ between any two vertices. To identify the automorphism group, we study $$\mathrm{GT}_\lambda $$ using combinatorial objects called $$ladder diagrams $$ and examine faces of co-dimension 2.

Sign matrix polytopes from Young tableaux

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,−1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.

A Cornucopia of Quasi-Yamanouchi Tableaux

Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and refine standard Young tableaux. They are closely tied to the descent set of standard Young tableaux and were introduced by Assaf and Searles to tighten Gessel's fundamental quasisymmetric expansion of Schur functions. The descent set and descent statistic of standard Young tableaux repeatedly prove themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make appearances in many ways outside of their original purpose. Some examples, which we present in this paper, include the Schur expansion of Jack polynomials, the decomposition of Foulkes characters, and the bigraded Frobenius image of the coinvariant algebra. While it would be nice to have a product formula enumeration of quasi-Yamanouchi tableaux in the way that semistandard and standard Young tableaux do, it has previously been shown by the author that there is little hope on that front. The goal of this paper is to address a handful of the numerous alternative enumerative approaches. In particular, we present enumerations of quasi-Yamanouchi tableaux using $q$-hit numbers, semistandard Young tableaux, weighted lattice paths, and symmetric polynomials, as well as the fundamental quasisymmetric and monomial quasisymmetric expansions of their Schur generating function.

Cyclic sieving and plethysm coefficients

A combinatorial expression for the coefficient of the Schur function s λ s_{\lambda } in the expansion of the plethysm p n / d d ∘ s μ p_{n/d}^d \circ s_{\mu } is given for all d d dividing n n for the cases in which n = 2 n=2 or λ \lambda is rectangular. In these cases, the coefficient ⟨ p n / d d ∘ s μ , s λ ⟩ \langle p_{n/d}^d \circ s_{\mu }, s_{\lambda } \rangle is shown to count, up to sign, the number of fixed points of an ⟨ s μ n , s λ ⟩ \langle s_{\mu }^n, s_{\lambda } \rangle -element set under the d d th power of an order- n n cyclic action. If n = 2 n=2 , the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if λ \lambda is rectangular, the action is a certain power of Schützenberger and Shimozono’s jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case n = 2 n=2 is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.

The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,…]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gröbner geometry of matrix Schubert varieties.