scholarly journals Symmetric products, linear representations and trace identities

2021 ◽  
Author(s):  
Francesco Vaccarino
Keyword(s):  
Commutative Ring ◽  
Characteristic Polynomial ◽  
Moduli Space ◽  
Symmetric Product ◽  
Infinite Field ◽  
Symmetric Tensors ◽  
Symmetric Products ◽  
Closed Immersion ◽  
Linear Representations ◽  
Affine Scheme

AbstractWe give the equations of the n-th symmetric product $$X^n/S_n$$ X n / S n of a flat affine scheme $$X=\mathrm {Spec}\,A$$ X = Spec A over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces.

QUANTUM COHOMOLOGIES OF SYMMETRIC PRODUCTS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250005 ◽  
Author(s):  
YONG SEUNG CHO
Keyword(s):  
Moduli Space ◽  
Product Space ◽  
Complex Projective Space ◽  
Projective Line ◽  
Symmetric Product ◽  
Witten Invariant ◽  
Symmetric Products ◽  
Complex Projective Line ◽  
Complex Projective ◽  
Invariant Properties

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov–Witten invariant and relative Gromov–Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line ℙ1, which is the k-dimensional complex projective space ℙk.

scholarly journals Indecomposable orthogonal invariants of several matrices over a field of positive characteristic

2020 ◽  
pp. 1-11
Author(s):  
Artem Lopatin
Keyword(s):  
Characteristic Polynomial ◽  
Orthogonal Group ◽  
General Linear Group ◽  
Positive Characteristic ◽  
The Other ◽  
Symmetric Matrices ◽  
Maximal Degree ◽  
Infinite Field ◽  
The Given ◽  
Field Of Positive Characteristic

We consider the algebra of invariants of [Formula: see text]-tuples of [Formula: see text] matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic [Formula: see text] different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic [Formula: see text] matrices. We establish that in case [Formula: see text] the maximal degree of indecomposable invariants tends to infinity as [Formula: see text] tends to infinity. In other words, there does not exist a constant [Formula: see text] such that it only depends on [Formula: see text] and the considered algebra of invariants is generated by elements of degree less than [Formula: see text] for any [Formula: see text]. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of [Formula: see text] the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

scholarly journals Discrepancy of Symmetric Products of Hypergraphs

10.37236/1066 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Benjamin Doerr ◽  
Michael Gnewuch ◽  
Nils Hebbinghaus
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Research Paper ◽  
Arithmetic Progressions ◽  
Symmetric Product ◽  
Upper And Lower Bounds ◽  
Symmetric Products ◽  
Cartesian Products ◽  
Better Than

For a hypergraph ${\cal H} = (V,{\cal E})$, its $d$–fold symmetric product is defined to be $\Delta^d {\cal H} = (V^d,\{E^d |E \in {\cal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${\rm disc}(\Delta^d {\cal H},2) \le {\rm disc}({\cal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${\rm disc}(\Delta^d {\cal H},c) = \Omega_d({\rm disc}({\cal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\cal H}^d$, which satisfies ${\rm disc}({\cal H}^d,c) = O_{c,d}({\rm disc}({\cal H},c)^d)$.

scholarly journals Conceptions of Topological Transitivity on Symmetric Products

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 61-80
Author(s):  
Franco Barragán ◽  
Sergio Macías ◽  
Anahí Rojas
Keyword(s):  
Positive Integer ◽  
Topological Space ◽  
Symmetric Product ◽  
Topological Transitivity ◽  
Symmetric Products ◽  
Weakly Mixing ◽  
The Relationship

Let X be a topological space. For any positive integer n , we consider the n -fold symmetric product of X , ℱ n ( X ), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : X → X , we consider the induced functions ℱ n ( ƒ ): ℱ n ( X ) → ℱ n ( X ). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ + -transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++ , semi-open and irreducible. In this paper we study the relationship between the following statements: ƒ ∈ M and ℱ n ( ƒ ) ∈ M .

The Homotopy Type of the Symmetric Products of Bouquets of Circles

2003 ◽  
Vol 14 (05) ◽  
pp. 489-497 ◽  
Author(s):  
Boon W. Ong
Keyword(s):  
Homotopy Type ◽  
Hyperplane Arrangement ◽  
Homotopy Equivalent ◽  
Symmetric Product ◽  
Symmetric Products

A wedge of k circles is homotopy equivalent to the plane with k points removed. By looking at [Formula: see text]∖{w1, …, wk} instead of ⋁k S1, we could view the symmetric product of a wedge of k circles as [Formula: see text], the complement space [3] of a hyperplane arrangement, [Formula: see text]. Applying a theorem of Hattori [2], we see that if k > n, the symmetric product of ⋁k S1 is, up to homotopy, the union of n-dimensional subtorii in a k-torus.

3-FOLD SYMMETRIC PRODUCTS OF CURVES AS HYPERBOLIC HYPERSURFACES IN ℙ4

2003 ◽  
Vol 14 (04) ◽  
pp. 413-436 ◽  
Author(s):  
CIRO CILIBERTO ◽  
MIKHAIL ZAIDENBERG
Keyword(s):  
Symmetric Product ◽  
Symmetric Products ◽  
Generic Projections

We construct new examples of Kobayashi hyperbolic hypersurfaces in ℙ4. They are generic projections of the triple symmetric product V = C (3) of a generic genus g ≥ 6 curve C, smoothly embedded in ℙ7.

scholarly journals Divisor Varieties of Symmetric Products

2021 ◽  
Author(s):  
John Sheridan
Keyword(s):  
Algebraic Curves ◽  
Linear Series ◽  
Symmetric Product ◽  
Fixed Degree ◽  
Symmetric Products ◽  
Projective Varieties ◽  
General Curve ◽  
Higher Dimensional ◽  
Detailed Picture

Abstract The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.

Is a typical bi-Perron algebraic unit a pseudo-Anosov dilatation?

2017 ◽  
Vol 39 (7) ◽  
pp. 1745-1750
Author(s):  
HYUNGRYUL BAIK ◽  
AHMAD RAFIQI ◽  
CHENXI WU
Keyword(s):  
Characteristic Polynomial ◽  
Moduli Space ◽  
Orientable Surface ◽  
Partial Answer ◽  
Closed Geodesics ◽  
Closed Orientable Surface ◽  
Abelian Differentials ◽  
Algebraic Unit ◽  
Almost All

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic units whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area-one abelian differentials for low-genus cases.

Symmetric products of the circle

1967 ◽  
Vol 63 (2) ◽  
pp. 349-352 ◽  
Author(s):  
H. R. Morton
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Cartesian Product ◽  
Symmetric Product ◽  
Manifolds With Boundary ◽  
Symmetric Products

The nth symmetric product of a topological space, X, is defined to be the quotient of the Cartesian product Xn by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S1, and to show that they are manifolds with boundary.

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