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Abstract: In this paper, we give an explicit formula for the Poincaré polynomial \(P_\lambda (x)\) for the Betti numbers of the Springer fibers over nilpotent elements in \(gl_n(\mathbb {C})\) of Jordan form \(\lambda =abc\) with \(a\ge b\ge c\ge 0\) at \(x=-1\) . In particular, we introduce \(\lambda \) -vacillating diagrams and show that \(P_{ab}(-1)\) is equal to the number of restricted Dyck paths. PubDate: 2022-05-11
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Abstract: In this paper, we introduce a determinant-like map \(\mathrm{det}^{S^3}\) and study some of its properties. For this, we define a graded vector space \(\Lambda ^{S^3}_V\) that has similar properties with the exterior algebra \(\Lambda _V\) and the exterior GSC-operad \(\Lambda ^{S^2}_V\) from Staic. When \(\mathrm{dim}(V_2)=2\) , we show that \(\mathrm{dim}_k(\Lambda ^{S^3}_{V_2}[6])=1\) , which gives the existence and uniqueness of the map \(\mathrm{det}^{S^3}\) . We also give an explicit formula for \(\mathrm{det}^{S^3}\) as a sum over certain 2-partitions of the complete hypergraph \(K_6^3\) . PubDate: 2022-05-11
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Abstract: Let \(L_n\) be a line graph with n edges and \({{\mathcal {F}}}(L_n)\) be the facet ideal of its matching complex. In this paper, we provide the irreducible decomposition of \({{\mathcal {F}}}(L_n)\) and some exact formulas for the projective dimension and regularity of \({{\mathcal {F}}}(L_n)\) . PubDate: 2022-05-11
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Abstract: A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the characteristic polynomial. Here, the conjecture is proved, and some of the other problems are solved, too. In the cases where also an old conjecture of Orlik on the integral monodromy holds, this has implications on the automorphism group of the Milnor lattice. The combinatorics used in the proof of the conjecture consists of tuples of orders on sets \(\{0,1,\ldots ,n\}\) with special properties and may be of independent interest. PubDate: 2022-05-09
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Abstract: Let the group G act transitively on the finite set \(\Omega \) , and let \(S \subseteq G\) be closed under taking inverses. The Schreier graph \(Sch(G \circlearrowleft \Omega ,S)\) is the graph with vertex set \(\Omega \) and edge set \(\{ (\omega ,\omega ^s) : \omega \in \Omega , s \in S \}\) . In this paper, we show that random Schreier graphs on \(C \log \Omega \) elements exhibit a (two-sided) spectral gap with high probability, magnifying a well-known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of G on \(\Omega \) , we give a lower bound on the number of elements which are necessary to provide a spectral gap. We use this method to estimate the spectral gap when G is nilpotent. PubDate: 2022-05-09
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Abstract: Abstract The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of \(\mathrm {PG}(2,q^n)\) , have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. The sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space \(\mathrm {PG}(3,q^n)\) have been classified in (Donati and Durante in J Algebr Comb 54:109–133, 2021). In this paper, we consider the four dimensional case and completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space \(\mathrm {PG}(4,q^n).\) As an application, we show that some of these sets are related to the Kantor’s ovoid and to the Tits’ ovoid of \(Q(4,q^n)\) and hence also to the Tits’ ovoid of \(\mathrm {PG}(3,q^n)\) . PubDate: 2022-05-03
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Abstract: A correction to this paper has been published: 10.1007/s10801-021-01048-4 PubDate: 2022-05-01
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Abstract: Abstract Following the definition of a root basis of an affine root system, we define a base of the root system R of an affine Lie superalgebra to be a linearly independent subset B of the linear span of R such that \(B\subseteq R\) and each root can be written as a linear combination of elements of B with integral coefficients such that either all coefficients are nonnegative or all coefficients are non-positive. Characterization and classification of bases of root systems of affine Lie algebras are known in the literature; in fact, up to \(\pm 1\) -multiple, each base of an affine root system is conjugate with the standard base under the Weyl group action. In the super case, the existence of those self-orthogonal roots which are not orthogonal to at least one other root, makes the situation more complicated. In this work, we give a complete characterization of bases of the root systems of twisted affine Lie superalgerbas with nontrivial odd part. We precisely describe and classify them. PubDate: 2022-05-01
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Abstract: Abstract Let \(S_n\) and \(A_{n}\) denote the symmetric and alternating group on the set \(\{1,\ldots ,n\},\) respectively. In this paper we are interested in the second largest eigenvalue \(\lambda _{2}(\Gamma )\) of the Cayley graph \(\Gamma =\mathrm{Cay}(G,H)\) over \(G=S_{n}\) or \(A_{n}\) for certain connecting sets H. Let \(1<k\le n\) and denote the set of all k-cycles in \(S_{n}\) by C(n, k). For \(H=C(n,n)\) we prove that \(\lambda _{2}(\Gamma )=(n-2)!\) (when n is even) and \(\lambda _{2}(\Gamma )=2(n-3)!\) (when n is odd). Further, for \(H=C(n,n-1)\) we have \(\lambda _{2}(\Gamma )=3(n-3)(n-5)!\) (when n is even) and \(\lambda _{2}(\Gamma )=2(n-2)(n-5) !\) (when n is odd). The case \(H=C(n,3)\) has been considered in Huang and Huang (J Algebraic Combin 50:99–111, 2019). Let \(1\le r<k<n\) and let \(C(n,k;r) \subseteq C(n,k)\) be set of all k-cycles in \(S_{n}\) which move all the points in the set \(\{1,2,\ldots ,r\}.\) That is to say, \(g=(i_{1},i_{2},\ldots ,i_{k})(i_{k+1})\dots (i_{n})\in C(n,k;r)\) if and only if \(\{1,2,\ldots ,r\}\subset \{i_{1},i_{2},\ldots ,i_{k}\}.\) Our main result concerns \(\lambda _{2}(\Gamma )\) , where \(\Gamma =\mathrm{Cay}(G,H)\) with \(H=C(n,k;r)\) with \(1\le r<k<n\) when \(G=S_{n}\) if k is even and \(G=A_{n}\) if k is odd. Here we observe that $$\begin{aligned} \lambda _{2}(\Gamma )\ge (k-2)! {n-r \atopwithdelims ()k-r} \frac{1}{n-r} \bi... PubDate: 2022-05-01
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Abstract: Abstract Let K be a field and let \(S=K[x_1,\ldots ,x_n]\) be a standard polynomial ring over a field K. We characterize the extremal Betti numbers, values as well as positions, of a t-spread strongly stable ideal of S. Our approach is constructive. Indeed, given some positive integers \(a_1,\dots ,a_r\) and some pairs of positive integers \((k_1,\ell _1),\ldots ,(k_r,\ell _r)\) , we are able to determine under which conditions there exists a t-spread strongly stable ideal I of S with \(\beta _{k_i, k_i+\ell _i}(I)=a_i\) , \(i=1, \ldots , r\) , as extremal Betti numbers, and then to construct it. PubDate: 2022-05-01
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Abstract: Abstract Sidon spaces can be used to characterize certain multiplicative properties of subspaces. They also have important applications in cyclic subspace codes and Sidon sets. In this paper, we give three new constructions of Sidon spaces. Our results can be applied to cyclic subspace codes. As a result, we obtain some new optimal cyclic subspace codes. PubDate: 2022-05-01
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Abstract: Abstract The directed power graph \(\vec {\mathcal {G}}( G)\) of a group G is the simple digraph with vertex set G in which \(x\rightarrow y\) if y is a power of x, and the power graph is the underlying simple graph \(\mathcal {G}( G)\) . In this paper, three versions of the definition of the power graph are discussed, and it is proved that the power graph by any of the three versions of the definition determines the other two up to isomorphism. It is also proved that if G is a torsion-free group of nilpotency class 2 and if H is a group such that \(\mathcal {G}( H)\cong \mathcal {G}( G)\) , then G and H have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina [9]. PubDate: 2022-05-01
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Abstract: Abstract In this paper, we compute the generating function for the Betti numbers of the Springer fibers over nilpotent elements in \(gl_n({\mathbb {C}})\) of Jordan form \((2^b,1^{a-b})\) , where \(a\ge b\ge 1\) . PubDate: 2022-05-01
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Abstract: Abstract Let \(g_1H_1,\ldots ,g_nH_n\) be cosets of subgroups \(H_1,\ldots ,H_n\) of a finite group G such that \(g_1H_1\cup \ldots \cup g_nH_n\ne G\) . We prove that \( g_1H_1\cup \ldots \cup g_nH_n \le \gamma _n G \) where \(\gamma _n<1\) is a constant depending only on n. In special cases, we show that \(\gamma _n=(2^n-1)/2^n\) is the best possible constant with this property and we conjecture that this is generally true. PubDate: 2022-05-01
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Abstract: Abstract We investigate several families of polynomials that are related to certain Euler type summation operators. Being integer valued at integral points, they satisfy combinatorial properties and nearby symmetries, due to triangle recursion relations involving squares of polynomials. Some of these interpolate the Delannoy numbers. The results are motivated by and strongly related to our study of irreducible Lie supermodules, where dimension polynomials in many cases show similar features. PubDate: 2022-05-01
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Abstract: Abstract Let G be a graph of order n, with vertex set \(V=\{v_{1},\dots ,v_{n}\}\) and adjacency matrix A. For a non-empty set S of vertices let \(\mathrm{e}=(x_{1}, \dots , x_{n})^\mathtt{T}\) be the characteristic vector of S, that is, \(x_{\ell }=1\) if \(v_{\ell }\in S\) and \(x_{\ell }=0\) otherwise. Then the \(n\times n\) matrix $$\begin{aligned} W^{S}:=\big [{\mathrm{e}}, A{\mathrm{e}}, A^{2}{\mathrm{e}},\dots ,A^{n-1}{\mathrm{e}}\big ] \end{aligned}$$ is the walk matrix of G for S. This term refers to the fact that in \(W^{S}\) the \(k{\mathrm{th}}\) entry in the row corresponding to \(v_{\ell }\) is the number of walks of length \(k-1\) from \(v_{\ell }\) to some vertex in S. Let \(\mu _{1},\dots , \mu _{s}\) be the distinct eigenvalues of A. For given S and characteristic vector \(\mathrm{e}\) , we re-arrange these eigenvalues in such a way that 1 $$\begin{aligned} {\mathrm{SD}}(S)\!:\mathrm{e}=\mathrm{e}_{1}+\mathrm{e}_{2}+\dots +\mathrm{e}_{r} \end{aligned}$$ for a certain \(r\le s,\) where the \(\mathrm{e}_{i}\) are eigenvectors of A for eigenvalue \(\mu _{i},\) for all \(1\le i\le r.\) We refer to (1) as the spectral decomposition of S, or more properly, of its characteristic vector \(\mathrm{e}.\) We show that the walk matrix \(W^{S}\) determines the spectral decomposition of S and vice versa. Explicit algorithms are given which establish this correspondence. In particular, we show that the number r of distinct eigenvectors that appear in (1) is equal to the rank of \(W^{S}.\) Various results can be derived from this theorem. We show that \(W^{S}\) determines the adjacency matrix of G if \(W^{S}\) has rank \(\ge n-1.\) Another application ... PubDate: 2022-05-01
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Abstract: Abstract Let G be a graph on the vertex set [n] and \(J_G\) the associated binomial edge ideal in the polynomial ring \(S=\mathbb {K}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]\) . In this paper, we investigate the depth of binomial edge ideals. More precisely, we first establish a combinatorial lower bound for the depth of \(S/J_G\) based on some graphical invariants of G. Next, we combinatorially characterize all binomial edge ideals \(J_G\) with \(\mathrm {depth} S/J_G=5\) . To achieve this goal, we associate a new poset \(\mathscr {M}_G\) with the binomial edge ideal of G and then elaborate some topological properties of certain subposets of \(\mathscr {M}_G\) in order to compute some local cohomology modules of \(S/J_G\) . PubDate: 2022-05-01
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Abstract: Abstract In this article, we investigate 2- \((v,k,\lambda )\) designs with \(\gcd (r,\lambda )=1\) admitting flag-transitive automorphism groups G. We prove that if G is an almost simple group, then such a design belongs to one of the eight infinite families of 2-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if \({\mathcal {D}}\) is a symmetric \((v,k,\lambda )\) design with \(\gcd (k,\lambda )=1\) admitting a flag-transitive automorphism group G, then either \(G\leqslant \mathrm {A}\Gamma \mathrm {L}_{1}(q)\) for some odd prime power q, or \({\mathcal {D}}\) is the point-hyperplane design or the unique Hadamard design with parameters (11, 5, 2). PubDate: 2022-05-01
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Abstract: Abstract Non-trivial 2- \((v,k,\lambda )\) designs, with \((r,\lambda )=1\) and \(\lambda >1\) , admitting a non-solvable flag-transitive automorphism group of affine type are classified. PubDate: 2022-05-01
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Abstract: Abstract We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph. PubDate: 2022-05-01
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