Standard Tableaux Research Articles

Kronecker powers of harmonics, polynomial rings, and generalized principal evaluations

Our main goal is to compute the decomposition of arbitrary Kronecker powers of the Harmonics of $$S_n$$ . To do this, we give a new way of decomposing the character for the action of $$S_n$$ on polynomial rings with k sets of n variables. There are two aspects to this decomposition. The first is algebraic, in which formulas can be given for certain restrictions from $$GL_n$$ to $$S_n$$ occurring in Schur-Weyl duality. The second is combinatorial. We give a generalization of the $${{\,\mathrm{{comaj}}\,}}$$ statistic on permutations which includes the $${{\,\mathrm{{comaj}}\,}}$$ statistic on standard tableaux. This statistic allows us to write a generalized principal evaluation for Schur functions and Gessel fundamental quasisymmetric functions.

Toward a conjecture of Pappas and Rapoport on a scheme attached to the symplectic group

We consider a closed subscheme V of the scheme of n × n skew-symmetric matrices. Over a field F of characteristic not 2, V is isomorphic to the scheme appearing in Pappas-Rapoport conjecture on local models of unitary Shimura varieties. With the additional assumption on we prove the coordinate ring of V has a basis consisting of products of pfaffians labeled by King’s symplectic standard tableaux with no odd-sized rows. When n is divisible by 4, the basis can be used to show that the coordinate ring of V is an integral domain, and this proves a special case of Pappas-Rapoport conjecture.

Transition Matrices Between Young's Natural and Seminormal Representations

We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.

Irreducible representations of the symmetric groups from slash homologies of p-complexes

In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called $p$-standard tableaux.

An insertion algorithm on multiset partitions with applications to diagram algebras

We generalize the Robinson–Schensted–Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson [15].

Random strict partitions and random shifted tableaux

We study asymptotics of random shifted Young diagrams which correspond to a given sequence of reducible projective representations of the symmetric groups. We show limit results (Law of Large Numbers and Central Limit Theorem) for their shapes, provided that the representation character ratios and their cumulants converge to zero at some prescribed speed. Our class of examples includes uniformly random shifted standard tableaux with prescribed shape as well as shifted tableaux generated by some natural combinatorial algorithms (such as shifted Robinson–Schensted–Knuth correspondence) applied to a random input.

Set-partition tableaux and representations of diagram algebras

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation.

A note on symmetry classes of tensors and Specht modules

We construct induced bases for critical orbital spaces indexed by standard tableaux. As a corollary we obtain an explicit isomorphism of C[Sm]-modules between Specht modules and critical orbital spaces which yields a realization of the complete set of complex irreducible representations of the symmetric group in terms of decomposable symmetrized tensors.

Simplex Parallelization in a Fully Hybrid Hardware Platform

The simplex method has been successfully used in solving linear programming (LP) problems for many years. Parallel approaches have also extensively been studied due to the intensive computations required, especially for the solution of large LP problems. Furthermore, the rapid proliferation of multicore CPU architectures as well as the computational power provided by the massive parallelism of modern GPUs have turned CPU / GPU collaboration models increasingly into focus over the last years for better performance. In this paper, a highly scalable implementation framework of the standard full tableau simplex method is first presented, over a hybrid parallel platform which consists of multiple multicore nodes interconnected via a high-speed communication network. The proposed approach is based on the combined use of MPI and OpenMP, adopting a suitable column-based distribution scheme for the simplex tableau. The parallelization framework is then extended in such a way that it can exploit concurrently the full power of the provided resources on a multicore single-node environment with a CUDA-enabled GPU (i.e. using the CPU cores and the GPU concurrently), based on a suitable hybrid multithreading/GPU offloading scheme with OpenMP and CUDA. The corresponding experimental results show that the hybrid MPI+OpenMP based parallelization scheme leads to particularly high speed-up and efficiency values, considerably better than in other competitive approaches, and scaling well even for very large / huge linear problems. Furthermore, the performance of the hybrid multithreading/GPU offloading scheme is clearly superior to both the OpenMP-only and the GPU-only based implementations in almost all cases, which validates the worth of using both resources concurrently. The most important, when it is used in combination with MPI in a multi-node (fully hybrid) environment, it leads to substantial improvements in the speedup achieved for large and very large LP problems.

On the Satisfiability of Quasi-Classical Description Logics

Though quasi-classical description logic (QCDL) can tolerate the inconsistency of description logic in reasoning, a knowledge base in QCDL possibly has no model. In this paper, we investigate the satisfiability of QCDL, namely, QC-coherency and QC-consistency and develop a tableau calculus, as a formal proof, to determine whether a knowledge base in QCDL is QC-consistent. To do so, we repair the standard tableau for DL by introducing several new expansion rules and defining a new closeness condition. Finally, we prove that this calculus is sound and complete. Based on this calculus, we implement an OWL paraconsistent reasoner called QC-OWL. Preliminary experiments show that QC-OWL is highly efficient in checking QC-consistency.

Bijections between oscillating tableaux and (semi)standard tableaux via growth diagrams

We prove that the number of oscillating tableaux of length n with at most k columns, starting at ∅ and ending at the one-column shape (1m), is equal to the number of standard Young tableaux of size n with m columns of odd length, all columns of length at most 2k. This refines a conjecture of Burrill, which it thereby establishes. We prove as well a “Knuth-type” extension stating a similar equi-enumeration result between generalised oscillating tableaux and semistandard tableaux.

Reasoning with Nominal Schemas through Absorption

Nominal schemas have recently been introduced as a new approach for the integration of DL-safe rules into the Description Logic framework. The efficient processing of knowledge bases with nominal schemas remains, however, challenging. We address this by extending the well-known optimisation of absorption as well as the standard tableau calculus to directly handle the (absorbed) nominal schema axioms. We implement the resulting extension of standard tableau calculi in the novel reasoning system Konclude and present further optimisations. In our empirical evaluation, we show the effect of these optimisations and we find that the proposed nominal schema handling performs well even when compared to (hyper)tableau systems with dedicated rule support.

The Combinatorics of $\mathsf{A_2}$-Webs

The nonelliptic $\mathsf{A_2}$-webs with $k$ "$+$''s on the top boundary and $3n-2k$ "$-$''s on the bottom boundary combinatorially model the space $\mathsf{Hom}_{\mathfrak{sl}_3}(\mathsf{V}^{\otimes (3n-2k)}, \mathsf{V}^{\otimes k})$ of $\mathfrak{sl}_3$-module maps on tensor powers of the natural 3-dimensional $\mathfrak{sl}_3$-module $\mathsf{V}$, and they have connections with the combinatorics ofSpringer varieties. Petersen, Pylyavskyy, and Rhodes showed that the set of such $\mathsf{A_2}$-webs and the set of semistandard tableaux of shape $(3^n)$ and type $\{1^2,\dots,k^2,k+1,\dots, 3n-k\}$ have the same cardinalities. In this work, we use the $\sf{m}$-diagrams introduced by Tymoczko and the Robinson-Schensted correspondence to construct an explicit bijection, different from the one given by Russell, between these two sets. In establishing our result, we show that the pair of standard tableaux constructed using the notion of path depth is the same as the pair constructed from applying the Robinson-Schensted correspondence to a $3\,2\,1$-avoiding permutation. We also obtain a bijection between such pairs of standard tableaux and Westbury's $\mathsf{A_2}$ flow diagrams.

Descent sets for symplectic groups

The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for tensor products of the defining representation of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor power of the defining representation of a symplectic group.

Charge on tableaux and the poset of k-shapes

A poset on a certain class of partitions known as k-shapes was introduced in [7] to provide a combinatorial rule for the expansion of a k−1-Schur function into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a k−1-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall–Littlewood polynomials into k-Schur functions), and show that it is compatible in the standard case with the weak bijection. In particular, we obtain that the usual charge of a standard tableau of size n is equal to the sum of the charges of its corresponding paths in the poset of k-shapes, for k=2,3,…,n.

Standard monomial theory for desingularized Richardson varieties in the flag variety GL(n)/B

We consider a desingularization Γ of a Richardson variety in the variety of complete flags, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. For any projective embedding of Z via a complete linear system, we construct a basis of the homogeneous coordinate ring of Γ inside Z, indexed by combinatorial objects which we call w 0 -standard tableaux.

Highly Scalable Parallelization of Standard Simplex Method on a Myrinet-Connected Cluster Platform

The simplex method has been successfully used in solving linear programming problems for many years. Parallel approaches have also extensively been studied due to the intensive computations required, especially for the solution of large linear problems (LPs). In this paper we present a highly scalable parallel implementation framework of the standard full tableau simplex method on a highly parallel (distributed memory) environment. Specifically, we have designed and implemented a suitable column distribution scheme as well as a row distribution scheme and we have entirely tested our implementations over a considerably powerful distributed platform (linux cluster with myrinet interface). We then compare our approaches (a) among each other for variable number of problem size (number of rows and columns) and (b) to other recent and valuable corresponding efforts in the literature. In most cases, the column distribution scheme performs quite/much better than the row distribution scheme. Moreover, both schemes (even the row distribution scheme over large-scale problems) lead to particularly high speedup and efficiency values, which are considerably better in all cases than the ones achieved in other similar research efforts and implementations. Moreover, we further evaluate our basic parallelization scheme over very large LPs in order to validate more reliably the high efficiency and scalability achieved.

Smooth components of Springer fibers

This article studies components of Springer fibers for 𝔤𝔩(n) that are associated to closed orbits of GL(p)×GL(q) on the flag variety of GL(n),n=p+q. These components occur in any Springer fiber. In contrast to the case of arbitrary components, these components are smooth varieties. Using results of Barchini and Zierau we show these components are iterated bundles and are stable under the action of a maximal torus of GL(n). We prove that if ℒ is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of ℒ to these components vanish. We derive some consequences of localization theorems in equivariant cohomology and K-theory, applied to these components. In the appendix we identify the tableaux corresponding to these components, under the bijective correspondence between components of Springer fibers for GL(n) and standard tableaux.

Homogeneous representations of Khovanov–Lauda Algebras

We construct irreducible graded representations of simply laced Khovanov–Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson–Proctor hook formula gives the dimensions of the homogeneous irreducible modules corresponding to straight shapes.

Algebraic tableau reasoning for the description logic [formula omitted

Semantic web applications based on the web ontology language (OWL) often require the use of numbers in class descriptions for expressing cardinality restrictions on properties or even classes. Some of these cardinalities are specified explicitly but quite a few are entailed and need to be discovered by reasoning procedures. Due to the description logic (DL) foundation of OWL those reasoning services are offered by DL reasoners which employ reasoning procedures that are arithmetically uninformed and substitute arithmetic reasoning by “don't know” non-determinism in order to cover all possible cases. This lack of information about arithmetic problems dramatically degrades the performance of DL reasoners in many cases, especially with ontologies relying on the use of nominals ( O ) and qualified cardinality restrictions ( Q ). In this article we present a new algebraic tableau reasoning procedure for the DL SHOQ that combines tableau procedures and algebraic methods, namely linear integer programming, to ensure arithmetically better informed reasoning procedures. SHOQ extends the standard DL ALC (which is equivalent to the multi-modal logic K m ) with transitive roles, role hierarchies, qualified cardinality restrictions, and nominals, and forms an expressive subset of the web ontology language OWL 2. Although the proposed algebraic tableau (in analogy to standard tableau) is still double exponential in the worst case, it deals with cardinalities in a very informed way due to its arithmetic component and can be considered as a novel foundation for informed reasoning procedures addressing cardinality restrictions.