In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system ( W , S ) : a partition of an element w is a subset 𝒫 ⊆ W such that the left inversion set of w is the disjoint union of the left inversion set of the elements in 𝒫 . Partitions of elements of W arises in the study of the Belkale-Kumar product on the cohomology H * ( X , ℤ ) , where X is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group 𝒮 n are also related to the Babington-Smith model in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone. Moreover, we state and discuss the conjecture that the number of right descents of w is the sum of the number of right descents of the elements of 𝒫 . In particular: (1) we state an equivalent conjecture in term of the number of atoms in the interval [ e , w ] R of the right weak order, with a word-metric flavour; (2) we give a direct proof that this conjecture holds in the cases of symmetric groups (type A ) and hyperoctahedral groups (type B ).
Some statements in the original publication relied on an implicit assumption. This corrigendum provides the necessary modifications when that assumption does not hold.
Using the connection between translation spreads of the generalized hexagon H ( q ) and linear sets (see Cardinali et al. in Eur J Comb 23:367–376, 2002; Lunardon and Polverino in J Algebraic Comb 18:255–262, 2003), the non-existence of 𝔽 q -translation spreads of H ( q 2 ) when p > 3 , q = p h , and q is large enough is proven. This answers to a question posed in [Marino and Polverino in J Algebraic Comb 42:725-744, 2015].
Jordan schemes generalize association schemes in a similar way as Jordan algebras generalize the associative ones. It is well-known that association schemes of maximal rank are in one-to-one correspondence with groups (so-called thin schemes). In this paper, we classify Jordan schemes of maximal rank-to-order ratio and show that regular Jordan schemes correspond to a special class of Moufang loops, known as Ring Alternative loops.
The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending ideas of Gillespie–Reimer-Berg, we use the RSK correspondence to give a positive interpretation of these counts in terms of the combinatorics of words.
The Geil-Matsumoto bound (GM bound) constrains the number of rational points on a curve over a finite field in terms of the Weierstrass semigroup of any of the points on the curve. For general numerical semigroups, the GM bound lacks a simple closed-form expression, making its computation a challenging problem. A closed formula has been obtained for the case when the semigroup is generated by two co-prime integers. In this work, for any numerical semigroup, we provide a closed formula for the GM bound in terms of the Apéry set of a nonzero element of the semigroup. In the case where the numerical semigroup is generated by consecutive integers n, n+1, …, n+t with ⌈n-1/2⌉≤ t ≤ n-1, we obtain a simple closed formula for the bound. We apply these results to obtain upper bounds on the number of rational points for algebraic curves over finite fields. In some cases, our bounds improve some well-known upper bounds on the number of rational points.
A MacMahon symmetric function is an invariant of the diagonal action of the symmetric group on power series in multiple alphabets of variables. We introduce an analogue of the chromatic symmetric function for vertex-weighted graphs, taking values in the MacMahon symmetric functions on two sets of variables, recording information about both cardinalities and weights of vertex sets. We prove that the chromatic symmetric MacMahon function of a tree determines the generating function for its vertex subsets by cardinality, weight, and the numbers of internal and external edges. This result generalizes the one for the unweighted case, first conjectured by Crew and proved independently by Aliste-Prieto–Martin–Wagner–Zamora and Liu–Tang.
Every maximum scattered linear set in PG(1, q5) is the projection of an Fq-subgeometry E of PG(4, q5) from a plane Gamma external to the secant variety to E [20]. The pair (Gamma, E) will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in PG(1, q5), namely those of pseudoregulus and LP type, have been characterized in the literature [6, 30]. Let (Gamma, E) be a projecting configuration for a maximum scattered linear set in PG(1, q5). Let sigma be a generator of G = P Gamma L(5, q5)Sigma, and A = Gamma boolean AND Gamma sigma 4, B = Gamma boolean AND Gamma sigma 3. If A and B are not both points, then the projected linear set is of pseudoregulus type [6]. Suppose that they are points. The rank of a point X is the vectorial dimension of the span of the orbit of X under the action of G. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points A and B has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that rkA = rkB = 4. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for q 5 25 no new maximum scattered linear set exists.
We introduce and study the Weingarten calculus for centered random permutation matrices in the symmetric group S_N. After presenting a formulation of the Weingarten calculus on the symmetric group, we derive a formula in the centered case, as well as a sign-respecting formula. Our investigations uncover the fact that a building block of this Weingarten calculus is Kummer's confluent hypergeometric function. It allows us to derive multiple algebraic properties of the Weingarten function and uniform estimate. These results shed a conceptual light on phenomena that take place regarding the algebraic and asymptotic behavior of moments of random permutations in the resolution of Bordenave and Bordenave-Collins of strong convergence. We obtain multiple new non-trivial estimates for moments of coefficients in centered moments.
The main goal of this paper is to settle a conceptual framework for cooperative game theory in which the notion of composition/aggregation of games is the defining structure. This is done via the mathematical theory of algebraic operads: we start by endowing the collection of all cooperative games with any number of players with an operad structure, and we show that it generalises all the previous notions of sums, products and compositions of games considered by Owen, Shapley, von Neumann and Morgenstern, and many others. Furthermore, we explicitly compute this operad in terms of generators and relations, showing that the Möbius transform map induces a canonical isomorphism between the operad of cooperative games and the operad that encodes commutative triassociative algebras. In other words, we prove that any cooperative game is a linear combination of iterated compositions of the 2-player bargaining game and the 2-player dictator games. We show that many interesting classes of games (simple, balanced, capacities a.k.a fuzzy measures and convex functions, totally monotone, etc) are stable under compositions, and thus form suboperads. In the convex case, this gives by the submodularity theorem a new operad structure on the family of all generalized permutahedra. Finally, we focus on how solution concepts in cooperative game theory behave under composition: we study the core of a composite and describe it in terms of the core of its components, and we give explicit formulas for the Shapley value and the Banzhaf index of a compound game.
For a finite subset X of the d-dimensional unit sphere, the harmonic strength T(X) of X is the set of ℓ∈ℕ such that ∑_x∈ X P(x)=0 for all harmonic polynomials P of homogeneous degree ℓ. We will study three exceptional finite groups of unit quaternions, called the binary tetrahedral group 2T of order 24, the octahedral group 2O of order 48, and the icosahedral group 2I of order 120, which can be viewed as a subset of the 3-dimensional unit sphere. For these three groups, we determine the harmonic strength and show the minimality and the uniqueness as spherical designs. In particular, the group 2O is unique as a minimal subset X of the 3-dimensional unit sphere with T(X)={22,14,10,6,4,2 }∪𝕆^+, where 𝕆^+ denotes the set of all positive odd integers. This result provides the first characterization of 2O from the spherical design viewpoint. For G∈{2T,2O,2I}, we consider the lattice 𝒪_G generated by G over R_G on which the group G acts on by multiplication, where R_2T=ℤ, R_2O=ℤ[√(2)], R_2I=ℤ[(1+√(5))/2] are the ring of integers. We introduce the spherical theta function θ_G,P(z) attached to the lattice 𝒪_G and a harmonic polynomial P of degree ℓ and prove that they are modular forms. By applying our results on the characterization of G as a spherical design, we determine the cases in which the ℂ-vector space spanned by all θ_G,P(z) of harmonic polynomials P of homogeneous degree ℓ has dimension zero–without relying on the theory of modular forms.
We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph K_2n. We show that they are special subsets of the association scheme obtained from the Gelfand pair (S_2n,S_2 ≀ S_n). This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra ℂ[S_2n] and using the representation theory of S_2n.
Pak-Robichaux recently introduced a signed puzzle rule for Schubert structure constants, which they use to show that sums γ_k(n) of these constants with a bounded number of inversions are polynomial. We give a different, conceptual proof of their theorem. Our argument computes the lead term of γ_k(n) and extends to all classical Lie types.
Let r 0, and let a and be partitions such that a1 r + 1. We present a combinatorial interpretation of the plethysm coefficient < s lambda, s & micro;[sr]>. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions a with a1 3, we find a combinatorial interpretation for the multiplicities of the irreducible C5n-submodules of the Schur module S lambda Cn, considered as an C5n-module.
A poset is called upper homogeneous, or "upho," if all of its principal order filters are isomorphic to the whole poset. In previous work of the first author, it was shown that each (finite-type N-graded) upho lattice has associated to it a finite graded lattice, called its core, which determines the rank generating function of the upho lattice. In that prior work the question of which finite graded lattices arise as cores was explored. Here, we study the question of in how many different ways a given finite graded lattice can be realized as the core of an upho lattice. We show that if the finite lattice has no nontrivial automorphisms, then it is the core of finitely many upho lattices. We also show that the number of ways a finite lattice can be realized as a core is unbounded, even when restricting to rank-two lattices. We end with a discussion of a potential algorithm for listing all the ways to realize a given finite lattice as a core.
A quasigroup is a pair (Q,*) where Q is a non-empty set and is a binary operation on Q such that for every (u, v) is an element of Q(2) there exists a unique (z, y) is an element of Q(2) such that u*x =v= y*u. Let q be an odd prime power, let. F-q denote the finite field of order q, and let R-q denote the set of non-zero squares in F-q. Let (a, b) is an element of F-q(2) be such that {ab (a - 1)(b - 1) R- q . Let Q(ub) denote the quadratic quasigroup (F-q. *(a,b)) where *(a,b) is defined by x(& lowast;a,b)y={x+a(y-x) if y-x is an element of R-q, x+b(y-x)otherwise. The operation table of a quadratic quasigroup is a quadratic Latin square. Recently, it has been determined exactly when two quadratic quasigroups are isomorphic and the automorphism group of any quadratic quasigroup has been determined. In this paper, we extend these results. We determine exactly when two quadratic quasigroups are isotopic and we determine the autotopism group of any quadratic quasigroup. In the process, we count the number of 2x2 subsquares in quadratic Latin squares.
We give a new formula for the Littlewood–Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel–Whitney rule. This gives an efficient way to compute generalized Littlewood–Richardson coefficients for Temperley–Lieb immanants of Jacobi–Trudi matrices. We will also show that our rule behaves well with Bender–Knuth involutions, recovering the symmetry of Littlewood–Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam–Postnikov–Pylyavskyy.
We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.
We provide a generalization of the Littlewood identity, both sides of which are related to alternating sign matrices. The classical Littlewood identity establishes a nice product formula for the sum of all Schur polynomials. Compared to the classical identity, Schur polynomials are replaced by so-called modified Robbins polynomials. These polynomials are a generalization of Schur polynomials and enumerate down-arrowed monotone triangles, and thus also alternating sign matrices. As an additional factor on the other side of the identity, we have a Pfaffian formula which we interpret in terms of the partition function of six-vertex model configurations corresponding to diagonally symmetric alternating sign matrices.
We prove a sharp representation stability result for graph complexes with a distinguished vertex, and prove that the chains realizing this sharp bound pass to non-trivial families of graph homology classes. This result may be interpreted as a higher genus generalization of Hersh and Reiner's stability bound for configuration spaces of points in odd dimensional Euclidean space.
