Natural Bijection Research Articles

The $(m, n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q, t$-symmetry

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area; skip; dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q; t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area; skip; dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q; t$-symmetry of $(m,n)$-rational $q; t$-Catalan polynomials for $m=3$.. Nous introduisons une nouvelle statistique, le skip, sur les chemins de $(3,n)$-Dyck rationnels et définissons le mot de rang marqué pour chaque chemin quand $n$ n’est pas un multiple de 3. Si un triplet valide de statistiques (aire, skip, dinv) est donné, nous avons un algorithme pour construire le mot de rang marqué correspondant au triplet. En considérant tous les triplets valides, nous donnons une formule explicite pour les polynômes de $q; t$-Catalan $(m,n)$- rationnels quand $m=3$. Enfin, il existe une bijection naturelle sur les triplets de statistiques (aire, skip, dinv) qui échange les statistiques aires et dinv en conservant le skip. Ainsi, nous prouvons la $q; t$-symétrie des polynômes de $q; t$-Catalan $(m,n)$-rationnels pour $m=3$..

Catalan numbers, binary trees, and pointed pseudotriangulations

We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed number of vertices with a fixed number of “marked” leaves or branching nodes, and the number of pointed pseudotriangulations of a certain class of planar point configurations.

Quotient closed subcategories of quiver representations

AbstractLet $Q$ be a finite quiver without oriented cycles, and let $k$ be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Weyl group $W_{Q}$ and the cofinite additive quotient closed subcategories of the category of finite dimensional right modules over $kQ$. We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to $Q$, which are also indexed by elements of $W_{Q}$.

Minimal length elements of extended affine Weyl groups

AbstractLet $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{{\mathcal{O}}}$ of any conjugacy class ${\mathcal{O}}$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79–94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_{{\mathcal{O}}}}$, where ${\mathcal{O}}$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H,H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404].

Invariants mod-2 and subgroups of [formula omitted] and [formula omitted

Let k be a field of characteristic different from 2 and 3. Let G be a simple group of type F4 or G2 defined over k. In this paper we discuss embeddings of connected simple algebraic groups of type A1 and A2 in G in terms of the mod-2 Galois cohomological invariants attached to these groups. We prove that k-groups of type F4 (resp. G2) arising from division algebras are generated by k-subgroups of type A2 (resp. A1) (see Theorems 3.11, 4.1). We derive a necessary and sufficient condition for an Albert algebra to have zero f5 invariant (Theorem 3.4). Further, for a k-group G of type F4, we derive a condition necessary for a k-group H of type A1 or A2 to embed in G over k. We prove that in order to embed H in G over k, the mod-2 invariant of H must divide f5(G) (see Section 2 and Remarks 2.8, 2.3 for definition of invariants). Along similar lines, we derive a condition for a k-group H of type A2 to embed in a k-group G of type G2. We prove that H embeds in G over k, if and only if f3(H)=f3(G) (Theorem 4.4). Next we derive a necessary and sufficient condition for a k-group H of type A1 to embed in a k-group G of type G2. If G=Aut(C) is a group of type G2 over k for an octonion algebra C over k, then this condition provides a natural bijection between k-conjugacy classes of involutions in G(k) and isometry classes of 2-fold Pfister divisors of the Pfister form nC, the norm form of the octonion algebra C (Section 4, Proposition 4.2).

On the number of cubic orders of bounded discriminant having automorphism group C3, and related problems

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed "lattice shape". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant $D$, the shape is equidistributed in the class group $\mathrm{Cl}_D$ of binary quadratic forms of discriminant $D$. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field.

Crossed products of [formula omitted]-algebras for singular actions

We consider group actions α:G→Aut(A) of topological groups G on C⁎-algebras A of the type which occurs in many physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a “crossed product host” in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action α:G→Aut(A). We prove a uniqueness theorem for crossed product hosts, and analyze existence conditions. We also present a number of examples where a crossed product host exists, but the usual crossed product does not. For actions where a crossed product host does not exist, we obtain a “maximal” invariant subalgebra for which a crossed product host exists. We further study the case of a discontinuous action α:G→Aut(A) of a locally compact group in detail.

Combinatorics of non-ambiguous trees

This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to give elegant combinatorial proofs of identities due to Carlitz, and to Ehrenborg and Steingrímsson. We also provide a hook formula to count the number of non-ambiguous trees with a given underlying tree. Finally, we use non-ambiguous trees to describe a very natural bijection between parallelogram polyominoes and binary trees.

Cluster equivalence and graded derived equivalence.

In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster-tilting objects in the generalized cluster categories: We first observe that cluster-tilting objects in generalized cluster categories are in natural bijection with cluster-tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that in general, if two algebras are cluster equivalent, then (under certain conditions) the algebras can be graded in such a way that the categories of graded modules are derived equivalent. To this end we introduce mutation of graded quivers with potential, and show that this notion reflects mutation in derived categories.

Presentations of Schützenberger groups of minimal subshifts

In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J-classes of free profinite semigroups. In this paper, the Sch\"utzenberger groups of such J-classes are investigated, in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for all non-periodic minimal subshifts associated with substitutions. It entails that it is decidable whether a finite group is a quotient of such a profinite group. As a further application, the Sch\"utzenberger group of the J-class corresponding to the Prouhet-Thue-Morse subshift is shown to admit a somewhat simpler presentation, from which it follows that it has rank three, and that it is non-free relatively to any pseudovariety of groups.

Enumeration of Fixed Points of an Involution on β(1, 0)-Trees

β(1, 0)-trees provide a convenient description of rooted non-separable planar maps. The involution h on β(1, 0)-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In this paper, we describe and enumerate fixed points of the involution h. Intriguingly, the fixed points are equinumerous with the fixed points under taking the dual map on rooted non-separable planar maps, even though the fixed points do not go to each other under the know (natural) bijection between the trees and the maps.

On [formula omitted]-stellated and [formula omitted]-stacked spheres

We introduce the class Σk(d) of k-stellated (combinatorial) spheres of dimension d (0≤k≤d+1) and compare and contrast it with the class Sk(d) (0≤k≤d) of k-stacked homology d-spheres. We have Σ1(d)=S1(d), and Σk(d)⊆Sk(d) for d≥2k−1. However, for each k≥2 there are k-stacked spheres which are not k-stellated. For d≤2k−2, the existence of k-stellated spheres which are not k-stacked remains an open question.We also consider the class Wk(d) (and Kk(d)) of simplicial complexes all whose vertex-links belong to Σk(d−1) (respectively, Sk(d−1)). Thus, Wk(d)⊆Kk(d) for d≥2k, while W1(d)=K1(d). Let K¯k(d) denote the class of d-dimensional complexes all whose vertex-links are k-stacked balls. We show that for d≥2k+2, there is a natural bijection M↦M¯ from Kk(d) onto K¯k(d+1) which is the inverse to the boundary map ∂:K¯k(d+1)→Kk(d).Finally, we complement the tightness results of our recent paper, Bagchi and Datta (2013) [5], by showing that, for any field F, an F-orientable (k+1)-neighbourly member of Wk(2k+1) is F-tight if and only if it is k-stacked.

Orders Induced by Segments in Floorplans and $(2 - 14 - 3, 3 - 41 - 2)$-Avoiding Permutations

A floorplan is a tiling of a rectangle by rectangles. There are natural ways to order the elements - rectangles and segments - of a floorplan. Ackerman, Barequet and Pinter studied a pair of orders induced by neighborhood relations between rectangles, and obtained a natural bijection between these pairs and $(2 - 41 - 3, 3 - 14 - 2)$-avoiding permutations, also known as (reduced) Baxter permutations.In the present paper, we first perform a similar study for a pair of orders induced by neighborhood relations between segments of a floorplan. We obtain a natural bijection between these pairs and another family of permutations, namely $(2 - 14 - 3, 3 - 41 - 2)$-avoiding permutations.Then, we investigate relations between the two kinds of pairs of orders - and, correspondingly, between $(2 - 41 - 3, 3 - 14 - 2)$- and $(2 - 14 - 3, 3 - 41 - 2)$-avoiding permutations. In particular, we prove that the superposition of both permutations gives a complete Baxter permutation (originally called $w$-admissible by Baxter and Joichi in the sixties). In other words, $(2 - 14 - 3, 3 - 41 - 2)$-avoiding permutations are the hidden part of complete Baxter permutations. We enumerate these permutations. To our knowledge, the characterization of these permutations in terms of forbidden patterns and their enumeration are both new results.Finally, we also study the special case of the so-called guillotine floorplans.

Maximal ideals and representations of twisted forms of algebras

Given a central simple algebra $\mathfrak{g}$ and a Galois extension of base rings $S/R$, we show that the maximal ideals of twisted $S/R$-forms of the algebra of currents $\mathfrak{g}(R)$ are in natural bijection with the maximal ideals of $R$. When $\mathfrak{g}$ is a Lie algebra, we use this to give a complete classification of the finite-dimensional simple modules over twisted forms of $\mathfrak{g}(R)$.

Cubic differentials and finite volume convex projective surfaces

We prove that there exists a natural bijection between the set of finite volume oriented convex projective surfaces with non-abelian fundamental group, and the set of finite volume hyperbolic Riemann surfaces endowed with a holomorphic cubic differential with poles of order at most 2 at the cusps.

Admissible Pictures and U 𝔮(𝔤𝔩(m, n))–Littlewood–Richardson Tableaux

We construct a natural bijection between the set of admissible pictures and the set of Uq(𝔤𝔩(m, n))–Littlewood–Richardson tableaux.

Weight Parameterization of Simple Modules for p-Solvable Groups

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

The Character Correspondences on π-Separable Groups

In this paper, we mainly consider the relationship between the complex irreducible characters of a π-separable group and the complex irreducible characters of its Hall π-subgroup. If a π-group S acts on a π-separable group G, let H be an S-invariant Hall π-subgroup of G and CNG(H)/H(S)=1. Then we construct a natural bijection from the set Lin S(H) onto the set Irr π′,S(G). Furthermore, we get a bijection from the linear characters of H onto Irr π′(G).

Parameterization of Irreducible Characters for p-Solvable Groups

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

The geodesic flow on a Riemannian supermanifold

We give a natural definition of geodesics on a Riemannian supermanifold (M,g) and extend the usual geodesic flow on T∗M associated to the underlying Riemannian manifold (M,g˜) to a geodesic “superflow” on T∗M. Integral curves of this flow turn out to be in natural bijection with geodesics on M. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds.