Semistandard Young Tableaux Research Articles

Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers

An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableaux of various shapes $\lambda$ and contents $\mu$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableaux that "standardize" to a fixed semistandard Young tableau. For $m$-row shapes $\lambda$ and standard content $\mu$, the total number of $k$-inverted standard Young tableaux of shape $\lambda$ is then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of "returns to ground" in those paths. In the rectangular specialization of $\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\lambda=2^n$.

Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

We introduce and study q -randomized Robinson–Schensted–Knuth (RSK) correspondences which interpolate between the classical ( q=0 ) and geometric q \nearrow1 ) RSK correspondences (the latter ones are sometimes also called tropical). For 0<q<1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q -Whittaker processes (which are t=0 versions of Macdonald processes of Borodin–Corwin [8]). We present four Markov dynamics which for q=0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries. Our new two-dimensional discrete time dynamics generalize and extend several known constructions. (1) The discrete time q -TASEPs studied by Borodin–Corwin [7] arise as one-dimensional marginals of our „column" dynamics. In a similar way, our“row" dynamics lead to discrete time q -PushTASEPs – new integrable particle systems in the Kardar–Parisi–Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q -PushASEP conjectured by Corwin–Petrov [23]. (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q -randomized column and row Robinson–Schensted correspondences introduced by O’Connell–Pei [59] and Borodin–Petrov [15], respectively. (3) In a scaling limit as q\nearrow1 , two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma (introduced by Seppäläinen [70] or strict-weak (introduced independently by O’Connell–Ortmann [58] and Corwin–Seppäläinen–Shen [25] directed random lattice polymers.

Characters for the Permutation Group of Degree n using Specht Module and Semi Standard Young Tableaux

For any partition, the corresponding Specht module is the sub module of permutation module spanned by the poly-tabloids. The Specht modules for a partition of a positive integer n form a complete list of irreducible representations of permutation group of degree n. The Semi-Standard Young Tableau (SSYT), on n -symbols are one type of combinatorial objects occur naturally in many computational problems in Science, Engineering and Technology, which have one-to-one correspondence with Gelfand–Tzetlin bases set of the Unitary group U (n).In this paper, we propose a method to construct character table of permutation group of degree n using Specht module and Semi Standard Young Tableaux. This method is illustrated with an example using a partition of degree 5 in permutation group S5.

An explicit bijection between semistandard tableaux and non-elliptic sl 3 webs

The sl 3 spider is a diagrammatic category used to study the representation theory of the quantum group U q (sl 3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov–Kuperberg observe that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and join of webs. This article builds on Tymoczko’s bijection to give a simple and explicit algorithm for constructing all non-elliptic sl 3 webs. As an application, we generalize results of Petersen–Pylyavskyy–Rhoades and Tymoczko proving that, for all non-elliptic sl 3 webs, rotation corresponds to jeu de taquin promotion and join corresponds to shuffling.

A Direct Way to Find the Right Key of a Semistandard Young Tableau

The right key of a semistandard Young tableau is a tool used to find Demazure characters for $${sl_n(\mathbb{C})}$$ . This paper gives methods to obtain the right and left keys by inspection of the semistandard Young tableau.

A non-crossing standard monomial theory

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand–Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets.

Parabosons, parafermions, and explicit representations of infinite-dimensional algebras

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labeled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. Alternatively, the basis vectors can be expressed as semi-standard Young tableaux.

Young tableaux and crystal base for U q (osp(1|2n))

Let U q (osp(1|2n)) be the quantized enveloping superalgebra corresponding to the Lie superalgebra osp(1|2n). In terms of semistandard Young tableaux satisfying some additional conditions, a realization of crystal graph of finite dimensional irreducible modules of U q (osp(1|2n)) is given. Also, the generalized Littlewood-Richardson rule for tensor product of crystal graphs is established.

The poset perspective on alternating sign matrices

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

An explicit construction of type A Demazure atoms

Demazure characters of type A, which are equivalent to key polynomials, have been decomposed by Lascoux and Schützenberger into standard bases. We prove that the resulting polynomials, which we call Demazure atoms, can be obtained from a certain specialization of nonsymmetric Macdonald polynomials. This combinatorial interpretation for Demazure atoms accelerates the computation of the right key associated to a semi-standard Young tableau. Utilizing a related construction, we provide a new combinatorial description of the key polynomials.

A universal enveloping for $L_\infty$-algebras.

For any $L_\infty$-algebra $L$ we construct an $A_\infty$-algebra structure on the symmetric coalgebra $Sym^*_c(L)$ and prove that this structure satisfies properties generalizing those of the usual universal enveloping algebra. These properties follow from an invariant contracting homotopy one the cobar construction of an exterior coalgebra and its relation to combinatorics of permutahedra and semistandard Young tableaux.

A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.

Truncations of Random Unitary Matrices and Young Tableaux

Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ For any $k \leq d,$ and any $k \times k$ submatrix $U_k$ of $U,$ we express the average value of $|{\rm Tr}(U_k)|^{2n}$ as a sum over partitions of $n$ with at most $k$ rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition, we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix, and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group.

Semistandard tableaux associated with generalized labellings of posets

We describe a correspondence between a family of labelled partially ordered sets and semistandard Young tableaux. Moreover, we define some operations among labelled posets which naturally correspond to operations among the associated semistandard Young tableaux.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

A Unified Approach to Combinatorial Formulas for Schubert Polynomials

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.

Joint Burke's Theorem and RSK Representation for a Queue and a Store

Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by $\mathcal{A}$ the arrival process and by $s$ the services. Assume the stability condition to be satisfied. Denote by $\mathcal{D}$ the departure process in equilibrium and by $r$ the time spent by the customers at the very back of the queue. We prove that $(\mathcal{D},r)$ has the same law as $(\mathcal{A},s)$ which is an extension of the classical Burke Theorem. In fact, $r$ can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.

Cones, crystals, and patterns

There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns. Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semi-simple algebraic groups.