Let F ${\mathbb F}$ double struck upper F be an algebraically closed field and G $G$ upper G be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups H
The limited dependence between the additive and the multiplicative structure of fields is in the background of a number of explicit constructions of various types of pseudorandom objects. In this direction we study the size of the intersection of (the additive) translates of fibers of the (multiplicative) norm function over finite fields. Besides extending earlier upper bounds, our main focus here is on obtaining lower bounds.
Let P $P$ upper P be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of Z $Z$ upper Z -polynomials associated with P $P$ upper P -kernels, providing a unified framework for various intersection cohomology Poincaré polynomials arising in diverse areas of mathematics. One of the problems posed by Proudfoot was to interpret the Z $Z$ upper Z -polynomial in a fundamental setting – namely, when P $P$ upper P is the lattice of faces of a convex polytope (or, more generally, an Eulerian poset). We resolve this problem by proving that the Z $Z$ upper Z -polynomial of any Eulerian poset coincides with the toric h $h$ h -polynomial of the poset of all (possibly empty) closed intervals of P $P$ upper P , ordered by reverse inclusion. Under suitable polyhedral conditions, this result identifies the Z $Z$ upper Z -polynomial of a polytope with the Poincaré polynomial of the intersection cohomology of an associated auxiliary polytope. We prove some results about the Chow polynomials of the poset of intervals of an Eulerian poset and relate them to the Veronese transforms of polynomials.
We prove that a smooth projective surface of degree d in P3 $\mathbb P^3$ double struck upper P cubed contains at most d2(d2−3d+3) $d^2(d^2-3d+3)$ d squared left parenthesis d squared minus 3 d plus 3 right parenthesis lines. We characterize the surfaces containing exactly d2(d2−3d+3) $d^2(d^2-3d+3)$ d squared left parenthesis d squared minus 3 d plus 3 right parenthesis lines: these occur only in prime characteristic p and, up to choice of projective coordinates, are cut out by equations of the form xpe+1+ype+1+zpe+1+wpe+1=0 $x^{p^{e}+1}+y^{p^{e}+1}+z^{p^{e}+1}+ w^{p^{e}+1} = 0$ x Superscript p Super Superscript e Superscript plus 1 Baseline plus y Superscript p Super Superscript e Superscript plus 1 Baseline plus z Superscript p Super Superscript e Superscript plus 1 Baseline plus w Superscript p Super Superscript e Superscript plus 1 Baseline equals 0 .
Let G be a transitive permutation group acting on Ω $\Omega $ normal upper Omega . In this paper, we introduce and study the parameter sep(G) $\mathrm {sep}(G)$ sep left parenthesis upper G right parenthesis , which denotes the size of the smallest set of points A such that, for every permutation g∈G $g\in G$ g element of upper G , A∩Ag $A \cap A^g$ upper A intersection upper A Superscript g is nonempty. In particular, we focus on deriving general bounds for arbitrary transitive groups, and on the asymptotic behaviour of certain families of primitive groups. We also provide a classification of transitive groups with sep(G) $\mathrm {sep}(G)$ sep left parenthesis upper G right parenthesis largest possible, namely with sep(G)=⌈(|Ω|+1)/2⌉ $\mathrm {sep}(G)=\lceil (|\Omega |+1) / 2 \rceil $ sep left parenthesis upper G right parenthesis equals left ceiling left parenthesis StartAbsoluteValue normal upper Omega EndAbsoluteValue plus 1 right parenthesis divided by 2 right ceiling .
Using Lück’s Chern character isomorphism we obtain a general formula in terms of centralisers for the p-adic Farrell–Tate K-theory of any discrete group G with a finite classifying space for proper actions. We apply this formula to Out(Fn) $\operatorname {\mathrm {Out}}(F_n)$ upper O u t left parenthesis upper F Subscript n Baseline right parenthesis . The case n=p+1 $n=p+1$ n equals p plus 1 turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order p element in Out(Fp+1) $\operatorname {\mathrm {Out}}(F_{p+1})$ upper O u t left parenthesis upper F Subscript p plus 1 Baseline right parenthesis which does not lift to an order p element in Aut(Fp+1) $\operatorname {\mathrm {Aut}}(F_{p+1})$ upper A u t left parenthesis upper F Subscript p plus 1 Baseline right parenthesis . We compute the rational cohomology of the centraliser of this element and as a consequence obtain a full calculation of the p-adic Farrell–Tate K-theory of Out(Fp+1) $\operatorname {\mathrm {Out}}(F_{p+1})$ upper O u t left parenthesis upper F Subscript p plus 1 Baseline right parenthesis for any prime p≥5 $p \geq 5$ p greater than or equals 5 . Our arguments provide an infinite family of Qp $\mathbb {Q}_p$ double struck upper Q Subscript p summands in K1(BOut(Fn))⊗ZQ $K^1(B\operatorname {\mathrm {Out}}(F_n)) \otimes _{\mathbb {Z}} \mathbb {Q}$ upper K Superscript 1 Baseline left parenthesis upper B upper O u t left parenthesis upper F Subscript n Baseline right parenthesis right parenthesis circled times Subscript double struck upper Z Baseline double struck upper Q , with no need for computer calculations. The smallest value of p to which this applies is p=11 $p=11$ p equals 11 . In this case we obtain a Q11 $\mathbb {Q}_{11}$ double struck upper Q 11 summand in K1(BOut(F12))⊗ZQ $K^1(B\operatorname {\mathrm {Out}}(F_{12})) \otimes _{\mathbb {Z}} \mathbb {Q}$ upper K Superscript 1 Baseline left parenthesis upper B upper O u t left parenthesis upper F 12 right parenthesis right parenthesis circled times Subscript double struck upper Z Baseline double struck upper Q .
The circular β $\beta $ beta ensemble for β=1,2 $\beta =1,2$ beta equals 1 comma 2 and 4 corresponds to circular orthogonal, unitary and symplectic ensemble respectively as introduced by Dyson. The statistical state of the eigenvalues is then a determinantal point process ( β=2 $\beta = 2$ beta equals 2 ) and Pfaffian point process ( β=1,4 $\beta = 1,4$ beta equals 1 comma 4 ). The explicit functional forms of the correlation kernels then imply that the general n-point correlation functions exhibit an asymptotic expansion in 1/N2 $1/N^2$ 1 divided by upper N squared , which moreover can be lifted to an asymptotic expansion in 1/N2 $1/N^2$ 1 divided by upper N squared for the spacing distributions and their generating function. We use σ $\sigma $ sigma -Painlevé characterisations to show that the functional form of the first correction is related to the leading term via a second derivative. In the case β=2 $\beta = 2$ beta equals 2 this finding has immediate consequences in interpreting the empirical Riemann zeros spacing distribution at large height, and that of their thinning. Explicit functional forms are used to show that the spectral form factors for β=1,2 $\beta =1,2$ beta equals 1 comma 2 and 4 also admit an asymptotic expansion in 1/N2 $1/N^2$ 1 divided by upper N squared . Differential relations are identified expressing the first and second correction in terms of the limiting functional form, and evidence is presented that they hold for general β $\beta $ beta . For even β $\beta $ beta it is proved that the two-point correlation function permits an asymptotic expansion in 1/N2 $1/N^2$ 1 divided by upper N squared , and moreover that the leading correction relates to the limiting functional form via a second derivative.
We introduce three families of vectors |λ―so⟩ $|\underline {\lambda }^{so}\rangle $ vertical bar lamda underbar Superscript s o Baseline right angle bracket , |λ―sp⟩ $|\underline {\lambda }^{sp}\rangle $ vertical bar lamda underbar Superscript s p Baseline right angle bracket and |λ―o⟩ $|\underline {\lambda }^{o}\rangle $ vertical bar lamda underbar Superscript o Baseline right angle bracket parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains [33, Section 7] and Betea [6, Section 3] are determinantal with respect to some explicit correlation kernels.
This is the fourth in a sequence of four papers, where we prove the arithmetic Siegel–Weil formula in co-rank 1 $1$ 1 for Kudla–Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel–Weil formula implies that degrees of Kudla–Rapoport arithmetic special 1 $1$ 1 -cycles are encoded in near-central first derivatives of unitary Eisenstein series Fourier coefficients.
We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on Zd $\mathbb Z^d$ double struck upper Z Superscript d , thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682–706, 2007]) to a nonlinear setting, and from the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1–45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrary-dimensional phase space, and the application of Bourgain’s geometric lemma in [Geom. Funct. Anal., 17(3):682–706, 2007].
We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.
Let $R o S$ be a cyclically pure map of Noetherian $\mathbb {Q}$ -algebras. In this paper, we show that if S has Du Bois singularities, then R has Du Bois singularities. Our result is new even when $R o S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R o S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb {C}$ , S has log canonical type singularities, and $K_R$ is Cartier, then R has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kov & aacute;cs and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underline {\Omega }<^>0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.
P & oacute;lya trees are unlabeled rooted trees on n vertices. This paper gives a new way to generate P & oacute;lya trees, that conjecturally is very efficient. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with "reality."Our method is an application of the Burnside process, alternating two steps: from a labeled rooted tree, produce a uniform permutation fixing it; and from a permutation, produce a uniform labeled rooted tree fixed by it. This last step is linked to a product formula, refining Cayley's, for the number of rooted labeled trees preserved by a given permutation.
Saturated fusion systems are categories modeling properties of conjugacy of p-elements in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group G generate a subnormal subgroup of G. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.
We prove that for polynomials $ f, g, h \in \mathbb {Z}[x] $ satisfying $ f = gh $ and $ f(0) \neq 0 $ , the $\ell _2$ -norm of the cofactor $ h $ is bounded by $$ \begin{align*} \left\Vert {h} \right\Vert{}_2\leq \sqrt{\frac{\left\Vert {g} \right\Vert{}_0}{\deg g}}\cdot \left\Vert {f} \right\Vert{}_1\cdot \left(\widetilde{O}\left( \frac{\left\Vert {g} \right\Vert{}_0<^>{2.5} \cdot \deg<^>2{f}}{\sqrt{\deg{g}}} \right) \right)<^>{\left\Vert {g} \right\Vert{}_0-1} \ , \end{align*} $$where $\left \Vert {g} \right \Vert {}_0$ is the number of nonzero coefficients of g (its sparsity). We also obtain similar results for polynomials over $\mathbb {C}$ . These bounds are an improvement over the bounds presented in an earlier conference version of this paper [NS24].This result significantly improves upon previously known exponential bounds (in $\deg {f}$ ) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $ h $ . This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials.In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and nonexact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, or even by a subquadratic function of $\deg {f}$ .
We derive exact formulas for the proportions of derangements and of derangements of p-power order in the affine classical groups AU(m)(q), ASp(2m)(q), AO(2m+1 )(q) and AO(2m )(+/-)(q), where p denotes the characteristic of the defining finite field. In the unitary case, the proofs of the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions lambda = (lambda(1 ), . . . , lambda(m)) into m parts, with lambda(1) > & centerdot; & centerdot; & centerdot; > lambda(m), such that either lambda(1) = 1 or lambda(k-1) > lambda(k )= k for some k is an element of {2, . . . , m}. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three q-polynomial identities conjectured by the author and later proved by Fulman and Stanton.
Let G be a simple algebraic group over an algebraically closed field $\Bbbk $ of positive characteristic. We consider the questions of when the tensor product of two simple G-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm {SL}_3(\Bbbk )$ and $G = \mathrm {Sp}_4(\Bbbk )$ .
Given a polynomial Sigma(nu)a(nu)X(nu)of degree < d, bounded by one on the unit disk, how large can a(0) + a(1) + ... + a(n) (n < d) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for d = 2(n + 1), but there does not seem to have been further progress on the question since. We study variations on exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that n is 'not too large' in terms of d. The latter is achieved via a 'quantitative' Enestr & ouml;m-Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums t(0)a(0) + ... + t(d-1)a(d-1) for arbitrary complex numbers t(0), ... , t(d-1). This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro's problem for some choices of n and d.
Let G be a simple algebraic group over an algebraically closed field $\Bbbk $ of positive characteristic. We consider the questions of when the tensor product of two simple G-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm {SL}_3(\Bbbk )$ and $G = \mathrm {Sp}_4(\Bbbk )$ .
