Journal of Algebraic Combinatorics最新文献

In this paper, we present results concerning the stabilizer (G_f) in ({{,mathrm{{GL}},}}(2,q^n)) of the subspace (U_f={(x,f(x)):xin mathbb {F}_{q^n}}), f(x) a scattered linearized polynomial in (mathbb {F}_{q^n}[x]). Each (G_f) contains the (q-1) maps ((x,y)mapsto (ax,ay)), (ain mathbb {F}_{q}^*). By virtue of the results of Beard (Duke Math J, 39:313–321, 1972) and Willett (Duke Math J 40(3):701–704, 1973), the matrices in (G_f) are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that (|G_f|>q-1) have a standard form of type (sum _{j=0}^{n/t-1}a_jx^{q^{s+jt}}) for some s and t such that ((s,t)=1), (t>1) a divisor of n; (ii) this standard form is essentially unique; (iii) for (n>2) and (q>3), the translation plane (mathcal {A}_f) associated with f(x) admits nontrivial affine homologies if and only if (|G_f|>q-1), and in that case those with axis through the origin form two groups of cardinality ((q^t-1)/(q-1)) that exchange axes and coaxes; (iv) no plane of type (mathcal {A}_f), f(x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in ({1,2,dots , n-1}). Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

Abstract

Let ({mathcal {D}}) be a non-trivial G-block-transitive 3-(v, k, 1) design, where (Tle G le textrm{Aut}(T)) for some finite non-abelian simple group T. It is proved that if T is a simple exceptional group of Lie type, then T is either the Suzuki group ({}^2B_2(q)) or (G_2(q)) . Furthermore, if (T={}^2B_2(q)) then the design ({mathcal {D}}) has parameters (v=q^2+1) and (k=q+1) , and so ({mathcal {D}}) is an inverse plane of order q, and if (T=G_2(q)) then the point stabilizer in T is either (textrm{SL}_3(q).2) or (textrm{SU}_3(q).2) , and the parameter k satisfies very restricted conditions.

Abstract

The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. In this paper, we determine the k-uniform hypertree with the minimal spectral radius among all k-uniform hypertrees with m edges and diameter d, where (3le dle m-1) .

This work is part of a research program to compute the Hochschild homology groups HH(_*({mathbb {C}}[x_1,ldots ,x_d]/(x_1,ldots ,x_d)^3;{mathbb {C}})) in the case (d = 2) through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism

provided by Larsen and Lindenstrauss. Here, (H_C^*) denotes Coxeter cohomology, (S_{i+j}) denotes the symmetric group on (i+j) letters, and V is the standard representation of (textrm{GL}_d({mathbb {C}})) on ({mathbb {C}}^d). We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of (S_n). In particular, the aforementioned tensor representation, and also several classes of irreducible representations of (S_n). Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial.

Abstract

A skew morphism of a finite group A is a permutation (varphi ) of A fixing the identity element and for which there is an integer-valued function (pi ) on A such that (varphi (ab)=varphi (a)varphi ^{pi (a)}(b)) for all (a, b in A) . A skew morphism (varphi ) of A is smooth if the associated power function (pi ) is constant on the orbits of (varphi ) , that is, (pi (varphi (a))equiv pi (a)pmod {|varphi |}) for all (ain A) . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if (n=2^en_1) , where (0 le e le 4) and (n_1) is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.

Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In Wang and Suzuki (Discrete Math 264:225–236, 2003), the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes and Doob graphs, respectively.

Let G be a finite group with identity e and ( H ne {e}) be a subgroup of G. The generalized non-coprime graph (varGamma _{G,H}) of ( G ) with respect to (H) is the simple undirected graph with (G setminus {e }) as the vertex set and two distinct vertices ( x ) and ( y) are adjacent if and only if (gcd (|x|,|y|) ne 1) and either (x in H) or (y in H), where |x| is the order of (xin G). In this paper, we study certain graph theoretical properties of generalized non-coprime graphs of finite groups, concentrating on cyclic groups. More specifically, we obtain necessary and sufficient conditions for the generalized non-coprime graph of a cyclic group to be in the class of stars, paths, triangle-free, complete bipartite, complete, split, claw-free, chordal or perfect graphs. Then we show that widening the class of groups to all finite nilpotent groups gives us no new graphs, but we give as an example of contrasting behaviour the class of EPPO groups (those in which all elements have prime power order). We conclude with a connection to the Gruenberg–Kegel graph.

Each connected graded, graded-commutative algebra A of finite type over a field (Bbbk ) of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When (A=Bbbk langle Delta rangle ) is the exterior Stanley–Reisner algebra associated to a finite simplicial complex (Delta ), we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group ((G,+)) is a bijection (f:Arightarrow B) between two finite subsets A, B of G satisfying (a+f(a)notin A) for all (ain A). A group G has the matching property if for every two finite subsets (A,B subset G) of the same size with (0 notin B), there exists a matching from A to B. In Losonczy (Adv Appl Math 20(3):385–391, 1998) it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group G, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.