Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).     

The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ 2 matrices

Let \(T_n(\mathbb {F})\) and \(UT_n(\mathbb {F})\) be the semigroups of all upper triangular \(n\times n\) matrices and all upper triangular \(n\times n\) matrices with 0s and/or 1s on the main diagonal over a field \(\mathbb {F}\) with \(\mathsf {char}(\mathbb {F})=0\), respectively. In this paper, we address the finite basis problem for \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) as involution semigroups under the skew transposition. By giving a sufficient condition under which an involution semigroup is nonfinitely based, we show that both \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety \(\mathsf {var} UT_2(\mathbb {F})\) generated by \(UT_2(\mathbb {F})\) and the involution monoid variety \(\mathsf {var} T_2(\mathbb {F})\) generated by \(T_2(\mathbb {F})\). Moreover, \(\mathsf {var} UT_2(\mathbb {F})\) cannot be defined within \(\mathsf {var} T_2(\mathbb {F})\) by any finite set of identities.     

Double ramification cycles on the moduli spaces of curves

Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.The cycle \(\mathsf{DR}_{g}(\mu,\nu)\) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for \(\mathsf{DR}_{g}(\mu,\nu)\) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.When \(\mu=\nu=\emptyset\), the formula for double ramification cycles expresses the top Chern class \(\lambda_{g}\) of the Hodge bundle of \(\overline {\mathcal{M}}_{g}\) as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

On counting subring-submodules of free modules over finite commutative frobenius rings

Let \(\texttt {R}\) be a finite commutative Frobenius ring and \(\texttt {S}\) a Galois extension of \(\texttt {R}\) of degree m. For positive integers k and \(k'\), we determine the number of free \(\texttt {S}\)-submodules \(\mathcal {B}\) of \(\texttt {S}^\ell \) with the property \(k=\texttt {rank}_\texttt {S}(\mathcal {B})\) and \(k'=\texttt {rank}_\texttt {R}(\mathcal {B}\cap \texttt {R}^\ell )\). This corrects the wrong result (Bill in Linear Algebr Appl 22:223–233, 1978, Theorem 6) which was given in the language of codes over finite fields.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Largest independent sets of certain regular subgraphs of the derangement graph

The derangement graph is the Cayley graph on the symmetric group \(\mathcal {S}_{n}\) whose generating set \(D_{n}\) is the set of permutations on \([n]=\{1, \ldots , n\}\) without any 1-cycle. For any fixed positive integer \(k \le n\), the Cayley graph generated by the subset of \(D_{n}\) consisting of permutations without any i-cycles for all \(1 \le i \le k\) is a regular subgraph of the derangement graph. In this paper, we determine the smallest eigenvalue of these subgraphs and show that the set of all the largest independent sets in these subgraphs is equal to the set of all the largest independent sets in the derangement graph, provided n is sufficiently large in terms of k.     

Dominating Cycles and Forbidden Pairs Containing $$P_{5}$$

A cycle C in a graph G is dominating if every edge of G is incident with at least one vertex of C. For a set \(\mathcal {H}\) of connected graphs, a graph G is said to be \(\mathcal {H}\)-free if G does not contain any member of \(\mathcal {H}\) as an induced subgraph. When \(|\mathcal {H}| = 2, \mathcal {H}\) is called a forbidden pair. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominating cycle and show the following two results: (i) Every 2-connected \(\{P_{5}, K_{4}^{-}\}\)-free graph contains a longest cycle which is a dominating cycle. (ii) Every 2-connected \(\{P_{5}, W^{*}\}\)-free graph contains a longest cycle which is a dominating cycle. Here \(P_{5}\) is the path of order \(5, K_{4}^{-}\) is the graph obtained from the complete graph of order 4 by removing one edge, and \(W^{*}\) is the graph obtained from two triangles and an edge by identifying one vertex in each.     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

Stringy K-theory and the Chern character

We construct two new G-equivariant rings: \(\mathcal{K}(X,G)\), called the stringy K-theory of the G-variety X, and \(\mathcal{H}(X,G)\), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack \(\mathcal{X}\), we also construct a new ring \(\mathsf{K}_{\mathrm{orb}}(\mathcal{X})\) called the full orbifold K-theory of \(\mathcal{X}\). We show that for a global quotient \(\mathcal{X} = [X/G]\), the ring of G-invariants \(K_{\mathrm{orb}}(\mathcal{X})\) of \(\mathcal{K}(X,G)\) is a subalgebra of \(\mathsf{K}_{\mathrm{orb}}([X/G])\) and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading.We prove that there is a ring isomorphism \(\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)\), which we call the stringy Chern character. We also show that there is a ring homomorphism \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\), which we call the orbifold Chern character, which induces an isomorphism \(Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\) when restricted to the sub-algebra \(K_{\mathrm{orb}}(\mathcal{X})\). Here \(H_{\mathrm{orb}}^\bullet(\mathcal{X})\) is the Chen–Ruan orbifold cohomology. We further show that \(\mathcal{C}\mathbf{h}\) and \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}\) preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.We further prove that \(\mathcal{H}(X,G)\) is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

Admissibility in Positive Logics

The paper studies admissibility of multiple-conclusion rules in positive logics. Using modification of a method employed by M. Wajsberg in the proof of the separation theorem, it is shown that the problem of admissibility of multiple-conclusion rules in the positive logics is equivalent to the problem of admissibility in intermediate logics defined by positive additional axioms. Moreover, a multiple-conclusion rule \(\mathsf {r}\) follows from a set of multiple-conclusion rules \(\mathsf {R}\) over a positive logic \(\mathsf {P}\) if and only if \(\mathsf {r}\) follows from \(\mathsf {R}\) over \(\mathbf {Int}+ \mathsf {P}\).     

Quadratic properties of least-squares solutions of linear matrix equations with statistical applications

Assume that a quadratic matrix-valued function \(\psi (X) = Q - X^{\prime }PX\) is given and let \(\mathcal{S} = \left\{ X\in {\mathbb R}^{n \times m} \, | \, \mathrm{trace}[\,(AX - B)^{\prime }(AX - B)\,] = \min \right\} \) be the set of all least-squares solutions of the linear matrix equation \(AX = B\). In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of \(\psi (X)\) subject to \(X \in {\mathcal S}\), and then derive from the formulas the analytic solutions of the two optimization problems \(\psi (X) =\max \) and \(\psi (X)= \min \) subject to \(X \in \mathcal{S}\) in the Löwner partial ordering. As applications, we present a variety of results on equalities and inequalities of the ordinary least squares estimators of unknown parameter vectors in general linear models.     

Convergence and Precise Asymptotics for Series Involving Self-normalized Sums

Let \(X, X_{1}, X_{2}, \ldots \) be i.i.d. random variables, and set \(S_{n}=X_{1}+\cdots +X_{n}\) and \( V_{n}^{2}=X_{1}^{2}+\cdots +X_{n}^{2}.\) Without any moment conditions on \(X\), assuming that \(\{S_{n}/V_{n}\}\) is tight, we establish convergence of series of the type (*) \(\sum \nolimits _{n}w_{n}P(\left| S_{n}\right| /V_{n}\ge \varepsilon b_{n}),\) \(\varepsilon >0.\) Then, assuming that \(X\) is symmetric and belongs to the domain of attraction of a stable law, and choosing \(w_{n}\) and \(b_{n}\) suitably\(,\) we derive the precise asymptotic behavior of the series (*) as \(\varepsilon \searrow 0. \)     

Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

In this article, we consider the following fractional Hamiltonian systems:

$$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$

where \(\alpha \in (1/2, 1)\), \(\lambda >0\) is a parameter, \(L\in C(\mathbb {R}, \mathbb {R}^{n\times n})\) and \(W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})\). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all \(t\in \mathbb {R}\), that is, \(L(t) \equiv 0\) is allowed to occur in some finite interval \(\mathbb {I}\) of \(\mathbb {R}\). Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on \(\mathbb {R} \setminus \mathbb {I}\) as \(\lambda \rightarrow \infty ,\) and converge to \(\tilde{u}\) in \(H^{\alpha }(\mathbb {R})\); here \(\tilde{u} \in E_{0}^{\alpha }\) is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval \(\mathbb {I}\). Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

     

Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary

We consider the problem

$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$

where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\ge 3,\) \(V_{\infty }>0,\) \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),\) \(\inf _{\mathbb {R}^{N}}V>-V_{\infty }\) and \(2<p<\frac{2N}{N-2}\). We assume V is periodic in the first m variables, and decays exponentially to zero in the remaining ones. We also assume that \(\Omega \) is periodic in the first m variables and has bounded complement in the other ones. Then, assuming that \(\Omega \) and V are invariant under some suitable group of symmetries on the last \(N-m\) coordinates of \(\mathbb {R}^{N}\), we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\)

     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.