Differential Posets Research Articles

Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

We present a general diagrammatic approach to the construction of efficient algorithms for computingthe Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to theconstruction of Fast Fourier Transform algorithms we make explicit use of the path algebra connection and work inthe setting of quivers. In this setting the complexity of an algorithm for computing a Fourier transform reduces to pathcounting in the Bratelli diagram, and we generalize Stanley's work on differential posets to provide such counts. Ourmethods give improved upper bounds for computing the Fourier transform for the general linear groups over finitefields, the classical Weyl groups, and homogeneous spaces of finite groups.

Path Counting and Rank Gaps in Differential Posets

We study the gaps Δpn between consecutive rank sizes in r-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller’s result that Δpn ≥ 1, which resolved a longstanding conjecture of Stanley, by showing that Δpn ≥ 2r. We also obtain stronger bounds in the case that the poset has many substructures called threads.

Dual Graded Graphs and Bratteli Diagrams of Towers of Groups

An $r$-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an $r$-dual graded graph. Miller and Reiner introduced a special case of these towers in order to study the Smith forms of the up and down maps in a differential poset. Agarwal and the author have also used these towers to compute critical groups of representations of groups appearing in the tower. In this paper I prove that when $r=1$ or $r$ is prime, wreath products of a fixed group with the symmetric groups are the only $r$-dual tower of groups, and conjecture that this is the case for general values of $r$. This implies that these wreath products are the only groups for which one can define an analog of the Robinson-Schensted bijection in terms of a growth rule in a dual graded graph.

Differential posets and restriction in critical groups

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a differential tower of groups, critical groups of certain representations are closely related to words of up-down maps in the associated differential poset. We use this to generalize an explicit formula for the critical group of the permutation representation of the symmetric group given by the second author, and to enumerate the factors in such critical groups.

Difference Operators for Partitions and Some Applications

Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the 2k-th power sum of hook lengths of partitions with size n is always a polynomial of n for any $${k \in \mathbb{N}}$$ . This conjecture was generalized and proved by Stanley (Ramanujan J. 23(1–3), 91–105 (2010)). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators D and $${D^{-}}$$ defined on functions of partitions. Even though the calculations for higher orders of D are extremely complex, we prove that several wellknown families of functions of partitions are annihilated by a power of the difference operator D. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants K r arise directly from the computation for a single partition $${\lambda}$$ , without the summation ranging over all partitions of size n.

Hochschild homology and cohomology of down–up algebras

We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi–Yau homogeneous down–up algebras. This family was defined by Benkart and Roby in [3] in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and some arguments similar to those used in [13] to compute the Hochschild cohomology of Yang–Mills algebras.

The Smith normal form of a matrix associated with Young’s lattice

We prove a conjecture of Miller and Reiner on the Smith normal form of the operator D U DU associated with a differential poset for the special case of Young’s lattice. Equivalently, this operator can be described as ∂ ∂ p 1 p 1 \frac {\partial }{\partial p_1}p_1 acting on homogeneous symmetric functions of degree n n .

A Note on Statistical Averages for Oscillating Tableaux

Oscillating tableaux are certain walks in Young's lattice of partitions; they generalize standard Young tableaux. The shape of an oscillating tableau is the last partition it visits and the length of an oscillating tableau is the number of steps it takes. We define a new statistic for oscillating tableaux that we call weight: the weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits. We show that the average weight of all oscillating tableaux of shape $\lambda$ and length $|\lambda|+2n$ (where $|\lambda|$ denotes the size of $\lambda$ and $n \in \mathbb{N}$) has a surprisingly simple formula: it is a quadratic polynomial in $|\lambda|$ and $n$. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm of Propp and Roby may lead to a more conceptual proof of this result and reveal a hidden symmetry in the set of perfect matchings.

Differential Posets have Strict Rank Growth: A Conjecture of Stanley

We establish strict growth for the rank function of an r-differential poset. We do so by exploiting the representation theoretic techniques developed by Reiner and the author (Order 26(3):197–228, 2009) for studying related Smith forms.

On the Rank Function of a Differential Poset

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.

Quantized Dual Graded Graphs

We study quantized dual graded graphs, which are graphs equipped with linear operators satisfying the relation $DU - qUD = rI$. We construct examples based upon: the Fibonacci differential poset, permutations, standard Young tableau, and plane binary trees.

Differential Posets and Smith Normal Forms

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU + tI and UD + tI have Smith normal forms over \({\mathbb Z}[t]\). In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young’s lattice Y and the r-differential Cartesian products Yr.

Formula omitted]-ribbon Fibonacci tableaux

We extend the notion of k -ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes k -colored permutations to pairs of k -ribbon Fibonacci tableaux of the same shape, and we demonstrate a color-to-spin property, similar to that described by Shimozono and White for ribbon tableaux. We describe a geometric interpretation of k -ribbon Fibonacci tableaux and use this interpretation to describe a notion of P equivalence for k -ribbon Fibonacci tableaux. In addition, we give an evacuation algorithm which relates the pair of k -ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of k -ribbon Fibonacci tableaux obtained using Fomin’s growth diagrams.

Signed differential posets and sign-imbalance

We study signed differential posets, a signed version of differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [T. Lam, Growth diagrams, domino insertion and sign-imbalance, J. Combin. Theory Ser. A 107 (2004) 87–115; A. Reifergerste, Permutation sign under the Robinson–Schensted–Knuth correspondence, Ann. Comb. 8 (2004) 103–112; J. Sjöstrand, On the sign-imbalance of partition shapes, J. Combin. Theory Ser. A 111 (2005) 190–203]. We show that these identities result from a signed differential poset structure on Young's lattice, and explain similar identities for Fibonacci shapes.

Combinatorial functional and differential equations applied to differential posets

We give combinatorial proofs of the primary results developed by Stanley for deriving enumerative properties of differential posets. In order to do this we extend the theory of combinatorial differential equations developed by Leroux and Viennot.

The Induced Subgraph Order on Unlabelled Graphs

A differential poset is a partially ordered set with raising and lowering operators $U$ and $D$ which satisfy the commutation relation $DU-UD=rI$ for some constant $r$. This notion may be generalized to deal with the case in which there exist sequences of constants $\{q_n\}_{n\geq 0}$ and $\{r_n\}_{n\geq 0}$ such that for any poset element $x$ of rank $n$, $DU(x) = q_n UD(x) + r_nx$. Here, we introduce natural raising and lowering operators such that the set of unlabelled graphs, ordered by $G\leq H$ if and only if $G$ is isomorphic to an induced subgraph of $H$, is a generalized differential poset with $q_n=2$ and $r_n = 2^n$. This allows one to apply a number of enumerative results regarding walk enumeration to the poset of induced subgraphs.

Truck Drivers, a Straw, and Two Glasses of Water

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsKevin IgaKevin Iga (kiga@pepperdine.edu) received his Ph.D. from Stanford University in 1998, studying four-dimensional topology under Ralph Cohen. Since then, he has taught at Pepperdine University (Malibu, CA 90263). His current research emphases include differential topology and the mathematics behind a physics theory called supersymmetry, but basically he works on any fun problem that comes his way.Kendra KillpatrickKendra Killpatrick (Kendra.Killpatrick@pepperdine.edu) received her Ph.D. from the University of Minnesota in 1998 in combinatorics. She spent three years after graduation as a post-doctoral student at Colorado State University and then began teaching at Pepperdine in 2002. Her main areas of research are enumerative combinatorics, especially permutation statistics and certain differential posets called Fibonacci posets. Outside mathematics, she loves to run marathons!

Evacuation and a geometric construction for Fibonacci tableaux

Tableaux have long been used to study combinatorial properties of permutations and multiset permutations. Discovered independently by Robinson and Schensted and generalized by Knuth, the Robinson–Schensted correspondence has provided a fundamental tool for relating permutations to tableaux. In 1963, Schützenberger defined a process called evacuation on standard tableaux which gives a relationship between the pairs of tableaux ( P , Q ) resulting from the Schensted correspondence for a permutation and both the reverse and the complement of that permutation. Viennot gave a geometric construction for the Schensted correspondence and Fomin described a generalization of the correspondence which provides a bijection between permutations and pairs of chains in Young's lattice. In 1975, Stanley defined a Fibonacci lattice and in 1988 he introduced the idea of a differential poset. Roby gave an insertion algorithm, analogous to the Schensted correspondence, for mapping a permutation to a pair of Fibonacci tableaux. The main results of this paper are to give an evacuation algorithm for the Fibonacci tableaux that is analogous to the evacuation algorithm on Young tableaux and to describe a geometric construction for the Fibonacci tableaux that is similar to Viennot's geometric construction for Young tableaux.

Centers of Down–Up Algebras

Down–up algebras which originated in the study of differential posets were recently defined and studied by Benkart and Roby. Benkart posed an open problem in her paper: to determine the centers of all down–up algebras. Here in this paper we completely solve this problem. As an application we also get the centers of homogenizations of down–up algebras.

Schensted Algorithms for Dual Graded Graphs

This paper is a sequel to l3r. We keep the notation and terminology and extend the numbering of sections, propositions, and formulae of l3r. The main result of this paper is a generalization of the Robinson-Schensted correspondence to the class of dual graded graphs introduced in l3r, This class extends the class of Y-graphs, or differential posets l22r, for which a generalized Schensted correspondence was constructed earlier in l2r. The main construction leads to unified bijective proofs of various identities related to path counting, including those obtained in l3r. It is also applied to permutation enumeration, including rook placements on Ferrers boards and enumeration of involutions. As particular cases of the general construction, we re-derive the classical algorithm of Robinson, Schensted, and Knuth l19, 12r, the Sagan-Stanley l18r, Sagan-Worley l16, 29r and Haiman's l11r algorithms and the author's algorithm for the Young-Fibonacci graph l2r. Some new applications are suggested. The rim hook correspondence of Stanton and White l23r and Viennot's bijection l28r are also special cases of the general construction of this paper. In l5r, the results of this paper and the previous paper l3r were presented in a form of extended abstract.