Set Of Standard Young Tableaux Research Articles

Two Partial Orders for Standard Young Tableaux

In this manuscript we show that two partial orders defined on the set of standard Young tableaux of shape $\alpha$ are equivalent. In fact, we give two proofs for the equivalence of the box order and the dominance order for tableaux. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of the cover relations. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.

Standard Rothe tableaux

Edelman and Greene constructed a bijection between the set of standard Young tableaux and the set of balanced Young tableaux of the same shape. Fomin, Greene, Reiner and Shimozono introduced the notion of balanced Rothe tableaux of a permutation w, and established a bijection between the set of balanced Rothe tableaux of w and the set of reduced words of w. In this paper, we introduce the notion of standard Rothe tableaux of w, which are tableaux obtained by labelling the cells of the Rothe diagram of w such that each row and each column is increasing. We show that the number of standard Rothe tableaux of w is smaller than or equal to the number of balanced Rothe tableaux of w, with equality if and only if w avoids the four patterns 2413, 2431, 3142 and 4132.

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.

Minimal Orbits of Promotion

We give a bijection between the symmetric group $S_n$, and the set of standard Young tableaux of rectangular shape $m^n$, $m \geq n$, that have order $n$ under jeu de taquin promotion.

Pattern Avoidance and Young Tableaux

This paper extends Lewis's bijection (J. Combin. Theorey Ser. A 118, 2011) to a bijection between a more general class $\mathcal{L}(n,k,I)$ of permutations and the set of standard Young tableaux of shape $\langle (k+1)^n\rangle$, so the cardinality\[|\mathcal{L}(n,k,I)|=f^{\langle (k+1)^n\rangle},\]is independent of the choice of $I\subseteq [n]$. As a consequence, we obtain some new combinatorial realizations and identities on Catalan numbers. In the end, we raise a problem on finding a bijection between $\mathcal{L}(n,k,I)$ and $\mathcal{L}(n,k,I')$ for distinct $I$ and $I'$.

Standard Young tableaux and colored Motzkin paths

In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the n-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length n. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most 2d+1 rows and the set of SYTs with at most 2d rows.

Alternating Permutations with Restrictions and Standard Young Tableaux

In this paper, we establish bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we show that 4123-avoiding up-down alternating permutations of length $2n+1$ are in one-to-one correspondence with standard Young tableaux of shape $(n+1,n,n-1)$, and 4123-avoiding up-down alternating permutations of length $2n$ are in bijection with shifted standard Young tableaux of shape $(n+2,n,n-2)$.

Generating Trees and Pattern Avoidance in Alternating Permutations

We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern $2143$. We use a generating tree approach to construct a recursive bijection between the set $A_{2n}(2143)$ of alternating permutations of length $2n$ avoiding $2143$ and the set of standard Young tableaux of shape $\langle n, n, n\rangle$, and between the set $A_{2n + 1}(2143)$ of alternating permutations of length $2n + 1$ avoiding $2143$ and the set of shifted standard Young tableaux of shape $\langle n + 2, n + 1, n\rangle$. We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.

Adjacency of Young tableaux and the Springer fibers

We connect different results about irreducible components of the Springer fibers of type A. Firstly, we show a relation between the Spaltenstein partition of the fibers and a total order \({\prec}\) on the set of standard Young tableaux. Next, using a result of Steinberg, we connect a work of the first author to the Robinson–Schensted map. We also perform the Spaltenstein study of the relative position of the Springer fibers and \({\mathbb{P}^1}\) -fibrations of the flag manifold. This leads us to consider the adjacency relation on the set of standard Young tableaux and to define oriented and labeled graphs with the standard Young tableaux as vertices. Using this adjacency relation, we describe some smooth irreducible components of the Springer fibers. Finally, we show that these graphs can be identified with some full subgraphs of the Bruhat graph.

A Relationship between the Major Index for Tableaux and the Charge Statistic for Permutations

The widely studied $q$-polynomial $f^{\lambda}(q)$, which specializes when $q=1$ to $f^{\lambda}$, the number of standard Young tableaux of shape $\lambda$, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation $S_q^{\lambda}$ of the finite general linear group $GL_n(q)$, it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index statistic on standard Young tableaux. Similarly, the $q$-polynomial $g^{\lambda}(q)$ has combinatorial interpretations as the $q$-multinomial coefficient, as the dimension of the permutation representation $M_q^{\lambda}$ of the general linear group $GL_n(q)$, and as the generating function for both the inversion statistic and the charge statistic on permutations in $W_{\lambda}$. It is a well known result that for $\lambda$ a partition of $n$, $dim(M_q^{\lambda}) = \Sigma_{\mu} K_{\mu \lambda} dim(S_q^{\mu})$, where the sum is over all partitions $\mu$ of $n$ and where the Kostka number $K_{\mu \lambda}$ gives the number of semistandard Young tableaux of shape $\mu$ and content $\lambda$. Thus $g^{\lambda}(q) - f^{\lambda}(q)$ is a $q$-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection $f$ from the set of standard Young tableaux of shape $\lambda$, $SYT(\lambda)$, to $W_{\lambda}$ such that $maj(T) = ch(f(T))$ for $T \in SYT(\lambda)$.

The Combinatorics of Orbital Varieties Closures of Nilpotent Order 2 in sl${}_n$

We consider two partial orders on the set of standard Young tableaux. The first one is induced to this set from the weak right order on symmetric group by Robinson-Schensted algorithm. The second one is induced to it from the dominance order on Young diagrams by considering a Young tableau as a chain of Young diagrams. We prove that these two orders of completely different nature coincide on the subset of Young tableaux with 2 columns or with 2 rows. This fact has very interesting geometric implications for orbital varieties of nilpotent order 2 in special linear algebra $sl_n.$

Components of the Springer fiber and domino tableaux

Consider a complex classical semisimple Lie group along with the set of its nilpotent coadjoint orbits. When the group is of type A, the set of orbital varieties contained in a given nilpotent orbit is described a set of standard Young tableaux. We parameterize both, the orbital varieties and the irreducible components of unipotent varieties in the other classical groups by sets of standard domino tableaux. The main tools are Spaltenstein's results on signed domino tableaux together with Garfinkle's operations on standard domino tableaux.

An Involution of the Variety of Flags Fixed by a Unipotent Linear Transformation

LetFbe the variety of full flags in2nand letG=GLn(3). Supposeuis a unipotent element inGand letFudenote the subvariety ofFconsisting of flags fixed byu. We define an involution, ρ, ofFu, and investigate its properties. It is known that ifuis in Jordan form, then there is a right cell,C, inSncanonically associated withu, and thatCindexes the irreducible components ofFu. In this paper, the elements inCare characterized in several ways. These characterizations are used to determine the action of ρ on the set of irreducible components ofFuand to show that ρ is the restriction toCof an involution of the generic Hecke algebra defined by Mathas. It is also shown that ρ intertwines maps fromFuto the set of standard Young tableaux defined by Spaltenstein and Steinberg.

Planarity Properties of the Schensted Correspondence

We give a natural decomposition of the set of standard Young tableaux of a given shape into intervals with respect to the weak Bruhat order; each class is completely determined by a partial order on letters which admits a remarkable planar representation.

Generalized quotients in Coxeter groups

For ( W W , S S ) a Coxeter group, we study sets of the form \[ W / V = { w ∈ W | l ( w v ) = l ( w ) + l ( v ) for all v ∈ V } , W/V = \{ w \in W|l(wv) = l(w) + l(v)\;{\text {for all}}\;v \in V\} , \] , where V ⊆ W V \subseteq W . Such sets W / V W/V , here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when V ⊆ S V \subseteq S (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in W / V W/V , for general V ⊆ W V \subseteq W , are lexicographically shellable. The Möbius function on W / V W/V under Bruhat order takes values in { − 1 , 0 , + 1 } \{ - 1,\,0,\, + 1\} . For finite groups W W , generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that W / V W/V is always a complete meet-semilattice and a convex order ideal as a subset of W W under weak order. Descent classes D I = { w ∈ W | l ( w s ) > l ( w ) ⇔ s ∈ I , for all s ∈ S } {D_I} = \{ w \in W|l(ws) > l(w) \Leftrightarrow s \in I,\;{\text {for all}}\;s \in S\} , I ⊆ S I \subseteq S , are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.