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Let j ≥ 3 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL(2, ℤ). Denote by λsymjf(n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L(s,symjf). In this paper, we are interested in the average behavior of λsymjf2(n) over arithmetic progressions. [ABSTRACT FROM AUTHOR]

We establish a variant for spheres of results obtained in [7], [3] for affine space. The principal result, that, if m is a power of 2 and k ≥ 1, then km continuous densities on the unit sphere in Rm+1 may be simultaneously bisected by a set of at most k hyperplanes through the origin, is essentially equivalent to the main theorem of Hubard and Karasev in [7]. But the methods used, involving Euler classes of vector bundles over symmetric powers of real projective spaces and an `orbifold' fixed point theorem for involutions, are substantially different from those in [7], [3]. [ABSTRACT FROM AUTHOR]

Three-dimensional manifolds carrying the structure of -valued coset topological groups originating from the Lie groups and are classified. [ABSTRACT FROM AUTHOR]

This paper treats the strict semi-stability of the symmetric powers S k E of a stable vector bundle E of rank 2 with even degree on a smooth projective curve C of genus g ≥ 2. The strict semi-stability of S 2 E is equivalent to the orthogonality of E or the existence of a bisection on the ruled surface ℙ C (E) whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of E with strictly semi-stable S 3 E. Moreover, it is shown that when S 2 E is stable, every symmetric power S k E is stable for all but a finite number of E in the moduli of stable vector bundles of rank  2 with fixed determinant of even degree on C. [ABSTRACT FROM AUTHOR]

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e ∈ sl (V) , we describe the Jordan normal form of e on the sl (V) -modules ∧ 2 (V) and S 2 (V). In the case where e is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element u ∈ SL (V) , the Jordan normal form of u on ∧ 2 (V) and S 2 (V). A recursive formula for the Jordan block sizes of u on ∧ 2 (V) was given by Gow and Laffey [J. Group Theory 9 (2006), 659–672]. We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S 2 (V). [ABSTRACT FROM AUTHOR]

Let n be a positive integer and λ be a partition of n. Let M λ be the Young permutation module labelled by λ. In this paper, we study symmetric and exterior powers of M λ in positive characteristic case. We determine the symmetric and exterior powers of M λ that are projective. All the indecomposable exterior powers of M λ are also classified. We then prove some results for indecomposable direct summands that have the largest complexity in direct sum decompositions of some symmetric and exterior powers of M λ. We end by parameterizing all the Scott modules that are isomorphic to direct summands of the symmetric or exterior square of M λ and determining their corresponding multiplicities explicitly. [ABSTRACT FROM AUTHOR]

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by λ f (n) . Let g(z) be another distinct primitive holomorphic cusp form of even integral weight ℓ for the full modular group. In this paper, we establish that the set { p | λ f (p j) < λ g (p j) } with 1 ≤ j ≤ 8 has analytic density at least 1 16 [ j + 1 2 ] 2 , and the set { p | λ f (p j) 2 < λ g (p j) 2 } with 1 ≤ j ≤ 4 has analytic density at least 1 4 j (j + 1) 2 . [ABSTRACT FROM AUTHOR]

Working over the prime field F p , the structure of the indecomposables Q ⁎ for the action of the algebra A (p) of Steenrod reduced powers on the symmetric power functors S ⁎ is studied by exploiting the theory of strict polynomial functors. In particular, working at the prime 2, representation stability is exhibited for certain related functors, leading to a conjectural representation stability theoretic description of quotients of Q ⁎ arising from the polynomial filtration of symmetric powers. [ABSTRACT FROM AUTHOR]

We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a multiplicity-free sum of power-sum symmetric functions. We describe a general framework in which such representations, and consequently such linear combinations of power sums, can be analysed. The conjugacy action for the symmetric group, and more generally for a large class of groups, is known to contain every irreducible. We find other representations of dimension n! with this property, including a twisted analogue of the conjugacy action. [ABSTRACT FROM AUTHOR]

Given a rank r stable bundle over a smooth irreducible projective curve C, there is an associated rank 2 r bundle over $$S^2(C),$$ the second symmetric power of C. In this article we study the slope (semi-)stability of this bundle. [ABSTRACT FROM AUTHOR]

ABSTRACT: Evolutionary theory predicts competition in nature yet altruistic and cooperative behaviour appears to reduce the ability to compete in order to help others compete better. This evolutionary puzzle is usually explained by kin selection where close relatives perform altruistic and cooperative acts to help each other and by reciprocity theory (i.e. direct, indirect and generalized reciprocity) among non‐kin. Here, it is proposed that the concepts of asymmetry and symmetry in power and dominance are critical if we are ever to resolve the puzzle of altruism and cooperation towards non‐kin. Asymmetry in power and dominance is likely to emerge under competition in nature as individuals strive to gain greater access to the scarce resources needed to survive and reproduce successfully. Yet asymmetric power presents serious problems for reciprocity theory in that a dominant individual faces a temptation to cheat in interactions with subordinates that is likely to far outweigh any individual selective benefits gained through reciprocal mechanisms. Furthermore, action taken by subordinates to deter non‐reciprocation by dominants is likely to prove prohibitively costly to their fitness, making successful enforcement of reciprocal mechanisms unlikely. It is also argued here that many apparently puzzling forms of cooperation observed in nature (e.g. cooperative breeding in which unrelated subordinates help dominants to breed) might be best explained by asymmetry in power and dominance. Once it is recognized that individuals in these cooperative interactions are subject to the constraints and opportunities imposed on them by asymmetric power then they can be seen as pursuing a ‘least bad’ strategy to promote individual fitness – one that is nevertheless consistent with evolutionary theory. The concept of symmetric power also provides important insights. It can inhibit reciprocal mechanisms in the sense that symmetric power makes it easier for a cheat to appropriate common resources while incurring fewer penalties. Nevertheless under certain restrictive conditions, symmetric power is seen as likely to promote direct reciprocity through ‘tit for tat’. [ABSTRACT FROM AUTHOR]

We introduce the symmetricity notions of symmetric h-monoidality, symmetroidality, and symmetric flatness. As shown in our paper [PS14a], these properties lie at the heart of the homotopy theory of colored symmetric operads and their algebras. In particular, the former property can be seen as the analog of Schwede and Shipley's monoid axiom for algebras over symmetric operads and allows one to equip categories of such algebras with model structures, whereas the latter ensures that weak equivalences of operads induce Quillen equivalences of categories of algebras. We discuss these properties for elementary model categories such as simplicial sets, simplicial presheaves, and chain complexes. Moreover, we provide powerful tools to promote these properties from such basic model categories to more involved ones, such as the stable model structure on symmetric spectra. This paper is also available at arXiv:1510.04969v3. [ABSTRACT FROM AUTHOR]

It is generally supposed that thin systems, including refracting surfaces and thin lenses, have powers that are necessarily symmetric. In other words they have powers which can be represented assymmetric dioptric power matrices and in the familar spherocylindrical form used in optometry and ophthalmology. This paper shows that this is not correct and that it is indeed possible for a thin system to have a power that is not symmetric and which cannot be expressed in spherocylindrical form. Thin systems of asymmetric power are illustratedby means of a thin lens that is modelled with small prisms and is chosen to have a dioptric power ma-trix that is antisymmetric. Similar models can be devised for a thin system whose dioptric power matrix is any 2 2 × matrix. Thus any power, symmetric, asymmetric or antisymmetric, is possible for a thin system. In this sense our understanding of the power of thin systems is now complete.

We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalized symmetric groups. We adapt the methods of Savitt and Stanley from [4] to obtain bounds both over the complex numbers and in prime characteristic. [ABSTRACT FROM PUBLISHER]

The convolution powers of a perverse sheaf on an abelian variety define an interesting family of branched local systems whose geometry is still poorly understood. We show that the generating series for their generic rank is a rational function of a very simple shape and that a similar result holds for the symmetric convolution powers. We also give formulae for other Schur functors in terms of characteristic classes on the dual abelian variety, and as an example we discuss the case of Prym-Tjurin varieties. [ABSTRACT FROM AUTHOR]

Let C be a smooth irreducible projective curve and E be a stable bundle of rank 2 on C . Then one can associate a rank 4 vector bundle F 2 ( E ) on S 2 ( C ) , the second symmetric power of C . Our goal in this article is to study semistability of this bundle. [ABSTRACT FROM AUTHOR]

We consider the asymptotic power dimension, i.e., the asymptotic dimension with control power function. It is proved that this dimension is invariant under the coarse bi-Hölder transformations. We also establish estimates for the power asymptotic dimension of the symmetric and hypersymmetric products. [ABSTRACT FROM AUTHOR]

Abstract: We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order. [Copyright &y& Elsevier]

We prove a result relating the stable generic Jordan types of the symmetric and exterior powers of the Heller translations of a module for a finite elementary abelian p-group. In the case of the trivial module, the stable generic Jordan types of the symmetric and exterior powers of its Heller translations are completely described. [ABSTRACT FROM AUTHOR]

In the previous article (Seyyedali, Duke Math. J. 153(3):573-605, ), we proved that slope stability of a holomorphic vector bundle E over a polarized manifold ( X, L) implies Chow stability of $(\mathbb{P}E^{*},\mathcal{O}_{\mathbb{P}E^{*}}(1)\otimes\pi^{*} L^{k})$ for k≫0 if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of 2 πc( L). In this article, using asymptotic expansions of the Bergman kernel on Sym E, we generalize the main theorem of Seyyedali (Duke Math. J. 153(3):573-605, ) to polarizations $(\mathbb{P}E^{*},\mathcal {O}_{\mathbb{P}E^{*}}(d)\otimes\pi^{*} L^{k})$ for k≫0, where d is a positive integer. [ABSTRACT FROM AUTHOR]

Abstract: Let K be a field of order p and let G be a cyclic group of order (). We explicitly decompose the symmetric powers of the indecomposable KG-module of dimension into indecomposable KG-modules. Using this result, for every odd prime p, we give a negative answer to the conjecture posed by Kouwenhoven [F.M. Kouwenhoven, The λ-structure of the Green rings of cyclic p-groups, Proc. Sympos. Pure Math. 47 (1987) 451–466]. [Copyright &y& Elsevier]

Let C be an irreducible smooth complex projective curve, and let E be an algebraic vector bundle of rank r on C. Associated to E, there are vector bundles $${{\mathcal F}_n(E)}$$ of rank nr on S( C), where S( C) is the n-th symmetric power of C. We prove the following: Let E and E be two semistable vector bundles on C, with genus $${(C)\, \geq\, 2}$$ . If $${{\mathcal F}_n(E_1)\,\simeq \, {\mathcal F}_n(E_2)}$$ for a fixed n, then $${E_1 \,\simeq\, E_2}$$ . [ABSTRACT FROM AUTHOR]

The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product ( S) by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ℂ P. We used the case d=2 of this fact to construct a 10-vertex triangulation of ℂ P earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of ( S), such that the $\operatorname{Sym}(3)$-action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$. Further, the $\operatorname{Sym}(3)$-action on $(S^{2})^{3}_{124}$ is 'good', so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ℂ P. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering ( S)→ℂ P. [ABSTRACT FROM AUTHOR]

We show how to derive new instances of the cyclic sieving phenomenon from old ones via elementary representation theory. Examples are given involving objects such as words, parking functions, finite fields, and graphs. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Green ring for a cyclic p-group C over a field K of prime characteristic p and determine the Adams operations in the case where n is not divisible by p. This gives information on the decomposition into indecomposables of exterior powers and symmetric powers of KC-modules. [Copyright &y& Elsevier]

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on Am such that (x1, ..., xm) ≡ (y1, ..., ym) if there exists a permutation σ on {1, ..., m} such that yσ(i) = xi for all i. Let A(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A(m) are said to be adjacent if (x1, ..., xm-1, a) ∈ X and (x1, ..., xm-1, b) ∈ Y for some x1, ..., xm-1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A(m) to B(n) that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors. [ABSTRACT FROM AUTHOR]

We study symmetric powers of classes of symmetric bilinear forms in the Witt–Grothendieck ring of a field of characteristic not equal to 2, and derive their basic properties and compute their classical invariants. We relate these to earlier results on exterior powers of such forms. [ABSTRACT FROM AUTHOR]

We study the degrees of the Abel--Jacobi mapping on hyperelliptic Riemann surfaces of arbitrary genus and the restrictions of the corresponding mappings to the symmetric powers of the real locus of the given Riemann surface. [ABSTRACT FROM AUTHOR]

We consider the degrees of the Abel--Jacobi maps for real hyperelliptic surfaces of genus $2$ and $3$. The restrictions of the maps to the symmetric square and the symmetric cube, respectively, of the real locus of the given Riemann surface are studied. [ABSTRACT FROM AUTHOR]

In this paper we give a modular interpretation of the k -th symmetric power L -function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of p = 2 and k odd give the precise 2-adic Newton polygon. We also give a p -adic modular interpretation of Dwork's unit root L -function of the Kloosterman family, and give the precise 2-adic Newton polygon when k is odd. In a previous paper, we gave an estimate for the q -adic Newton polygon of the symmetric power L -function of the Kloosterman family when p ≥ 5. We discuss how this restriction on primes was not needed, and also provide an improved estimate. [ABSTRACT FROM AUTHOR]

We define, for each subset S of the set P of primes, an S n -module L i e n S with interesting properties. L i e n ∅ is the well-known representation L i e n of S n afforded by the free Lie algebra, while L i e n P is the module C o n j n of the conjugacy action of S n on n -cycles. For arbitrary S the module L i e n S interpolates between the representations L i e n and C o n j n. We consider the symmetric and exterior powers of L i e n S. These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher L i e n S modules can be elegantly expressed as a multiplicity-free sum of power sums. In particular this establishes the Schur positivity of new classes of sums of power sums. More generally, for each nonempty subset T of positive integers we define a sequence of symmetric functions f n T of homogeneous degree n. We show that the series ∑ λ , λ i ∈ T p λ can be expressed as symmetrised powers of the functions f n T , analogous to the higher Lie modules first defined by Thrall. This in turn allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones. We also uncover some curious plethystic relationships between f n T , the conjugacy action and the Lie representation. [ABSTRACT FROM AUTHOR]