Nth Symmetric Power Research Articles

The structure of symmetric tensor powers of composition algebras

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.

Elliptic Double Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C<sub>n</sub> version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

On the pre-λ-ring structure on the Grothendieck ring of stacks and the power structures over it

We describe a pre-λ-structure on the Grothendieck ring of stacks (originally studied by Torsten Ekedahl) and the corresponding power structures over it, discuss some of their properties and give some explicit formulae for the Kapranov zeta-function for some stacks. In particular, we show that the nth symmetric power of the class of the classifying stack BGL(1) of the group GL(1) coincides, up to a power of the class 핃 of the affine line, with the class of the classifying stack BGL(n).

Divisors on rational normal scrolls

Let A be the homogeneous coordinate ring of a rational normal scroll. The ring A is equal to the quotient of a polynomial ring S by the ideal generated by the two by two minors of a scroll matrix ψ with two rows and ℓ catalecticant blocks. The class group of A is cyclic, and is infinite provided ℓ is at least two. One generator of the class group is [ J ] , where J is the ideal of A generated by the entries of the first column of ψ. The positive powers of J are well-understood; if ℓ is at least two, then the nth ordinary power, the nth symmetric power, and the nth symbolic power coincide and therefore all three nth powers are resolved by a generalized Eagon–Northcott complex. The inverse of [ J ] in the class group of A is [ K ] , where K is the ideal generated by the entries of the first row of ψ. We study the positive powers of [ K ] . We obtain a minimal generating set and a Gröbner basis for the preimage in S of the symbolic power K ( n ) . We describe a filtration of K ( n ) in which all of the factors are Cohen–Macaulay S-modules resolved by generalized Eagon–Northcott complexes. We use this filtration to describe the modules in a finely graded resolution of K ( n ) by free S-modules. We calculate the regularity of the graded S-module K ( n ) and we show that the symbolic Rees ring of K is Noetherian.

On the injectivity of blowing-up ring morphisms

Let I = ( x 1 , … , x r ) be a finitely generated ideal in a commutative ring R and let n ⩾ 2 be an integer. Let α I , n : S n ( I ) → I n be the canonical morphism from the nth symmetric power of I onto the nth power of I. Recently Tchernev asked for when α I , n being an isomorphism implies that α I , p is an isomorphism for each 2 ⩽ p ⩽ n . We give an affirmative answer provided that the ideal J = ( x 1 , … , x r − 1 ) verifies that α J , p : S p ( J ) → J p is an isomorphism for all 2 ⩽ p ⩽ n . In addition, for every n ⩾ 2 , we give an example of an ideal I such that α I , p is an isomorphism for all p ⩾ n + 1 and α I , n is not.

Asymptotic prime divisors of torsion-free symmetric powers of modules

Let R be a Noetherian ring, F : = R r and M ⊆ F a submodule of rank r. Let A ∗ ¯ ( M ) denote the stable value of Ass ( F n / M n ¯ ) , for n large, where F n is the nth symmetric power of F n and M n is the image of the nth symmetric power of M in F n . We provide a number of characterizations for a prime ideal to belong to A ∗ ¯ ( M ) . We also show that A ∗ ¯ ( M ) ⊆ A ∗ ( M ) , where A ∗ ( M ) denotes the stable value of Ass ( F n / M n ) .

Quantization of minimal resolutions of Kleinian singularities

In this paper we prove an analogue of a recent result of Gordon and Stafford that relates the representation theory of certain noncommutative deformations of the coordinate ring of the nth symmetric power of C 2 with the geometry of the Hilbert scheme of n points in C 2 through the formalism of Z -algebras. Our work produces, for every regular noncommutative deformation O λ of a Kleinian singularity X = C 2 / Γ , as defined by Crawley-Boevey and Holland, a filtered Z -algebra which is Morita equivalent to O λ , such that the associated graded Z -algebra is Morita equivalent to the minimal resolution of X. The construction uses the description of the algebras O λ as quantum Hamiltonian reductions, due to Holland, and a GIT construction of minimal resolutions of X, due to Cassens and Slodowy.

Rationality criteria for motivic zeta functions

The zeta function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta function of a surface is rational if and only if the Kodaira dimension of the surface is negative.

Microcanonical thermodynamic properties of helium nanodroplets

The density of states and other thermodynamic functions of helium nanodroplets are calculated for a microcanonical ensemble with both energy and total angular momentum treated as conserved quantum numbers. These functions allow angular momentum conserving evaporative cooling simulations. As part of this project, a recursion relationship is derived for the reduction to irreducible representations of the nth symmetric power of the irreducible representations of the rotation group. These give the distribution of total angular momentum states generated by putting multiple quanta into a ripplon or phonon mode of the droplet, each of which is characterized by a angular momentum quantum number.

ABELIAN FIBRATIONS AND RATIONAL POINTS ON SYMMETRIC PRODUCTS

Given a variety over a number field, are its rational points potentially dense, i.e. does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an Abelian fibration over ℙN. It is an interesting geometric problem to find the smallest N with this property.

On the Depth of the Invariants of the Symmetric Power Representations of SL2(Fp)

We study the depth of the ring of invariants of SL2(Fp) acting on the nth symmetric power of the natural two-dimensional representation for n<p. These symmetric power representations are the irreducible representations of SL2(Fp) over Fp. We prove that, when the greatest common divisor of p−1 and n is less than or equal to 2, the depth of the ring of invariants is 3. We also prove that the depth is 3 for n=3, p≠7 and n=4, p≠5. However, for n=3, p=7 the depth is 4 and for n=4, p=5 the depth is 5. In these two exceptional cases, the ring of invariants is Cohen–Macaulay.

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let S be a polynomial ring over a field. For a graded S-module generated in degree at most P, the Castelnuovo-Mumford regularity of each of (i) its nth symmetric power, (ii) its nth torsion-free symmetric power and (iii) the integral closure of its nth torsion-free symmetric power is bounded above by a linear function in n with leading coefficient at most P. For a graded ideal I of S, the regularity of I' is given by a linear function of n for all sufficiently large n. The leading coefficient of this function is identified. Let S = k[xl,... , Xd] be a polynomial ring over a field k with its usual grading, i.e., each xi has degree 1, and let m denote the maximal graded ideal of S. Let N be a finitely generated non-zero graded S-module. The Castelnuovo-Mumford regularity of N, denoted reg(N), is defined to be the least integer m so that, for every j, the Jth syzygy of N is generated in degrees F -> o N --> 0 where Fi = 1 S(-aij) for some integers ai -which we will refer to as the twists of Fi. Then, reg(N) < maxij {aij i} with equality holding if the resolution is minimal. For other equivalent definitions and properties of this invariant, see [Snb]. For a graded ideal I in S, the behaviour of the regularity of I' as a function of n has been of some interest. If I defines a smooth complex projective variety, it is shown in [BrtEinLzr, Proposition 1] using the Kawamata-Viehweg vanishing theorem that reg(In) < Pn + Q where P is the maximal degree of a minimal generator of I and Q is a constant expressed in terms of the degrees of generators of I. In [GrmGmgPtt, Theorem 1.1] and in [Chn, Theorem 1] it is shown that if dim(R/I) < 1, then reg(In) < n. reg(I) for all n E N. In [Chn, Conjecture 1], this is conjectured to be true for an arbitrary graded ideal. Supporting this conjecture is the result of [Swn, Theorem 3.6] that reg(In) < Pn for some constant P and for all n E N. The method of proof makes it difficult to explicitly identify such a constant. For monomial ideals, such a P is explicitly calculated in [SmtSwn, Theorem 3.1] and improved upon in [HoaTrn, Corollary 3.2]. We show that with S and N as above, the regularity of Symn(N) and related modules is bounded above by a linear function of n with leading coefficient at Received by the editors October 28, 1997 and, in revised form, April 15, 1998. 1991 Mathematics Subject Classification. Primary 13D02; Secondary 13D40. (?1999 American Mathematical Society

Rational equivalence of 0-cycles on surfaces

We will consider in this note 0-cycles on a complete non-singular algebraic surface F over the field C o f complex numbers. We will use the language o f schemes, and every scheme will be assumed separated and of finite type over C. In a very extensive set of papers, Severi set up and investigated the concept of rational equivalence (cf. [2], [3], [4], [5] among many others). It is not however very easy to find a precise definition in Seven's work, and there was a good deal of discussion on this point a t the International Congress o f 1954. A much more elementary approach was worked out by Chevalley in his seminar Anneaux de Chow [1]. For 0cycles on F, the most elementary definition is this: Let E„—group of permutations on n letters. Let SF= FlE„, the nth symmetric power of F {Al A effective 0-cycle of degree n on F } . SS Set