Abstract: Let G be a reductive linear algebraic group defined over an algebraically closed base field k of characteristic zero. A G-variety X is an algebraic variety with a regular action of G, defined over k. An affine G-variety Y is called stable if points of Y in general position have closed G-orbits. Our main result is a necessary and sufficient condition for a G-variety X to have a stable affine birational model.
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Abstract: We study quadratic forms that can occur as trace forms of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that any such form is a scaled Pfister form. We prove this conjecture and classify the finite groups G which admit a G-Galois extension L/K with a non-hyperbolic trace form. We also give several applications of these results.
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Abstract: This paper was inspired by the work of Herbert Hauptman, a leading figure in X-ray crystallography and a 1985 Nobel laureate in chemistry. From a mathematician's point of view, the situation is roughly as follows. To determine the structure of a physical crystal, one needs to know certain quantities, called ``phases". In practice, these cannot be measured directly; however, one can measure another set of quantities called ``observables" or ``magnitudes". The question then becomes: if the ``magnitudes" are known, can one recover the ``phases", and if so, what is the most efficient way to carry out the computations? Hauptman and his collaborators pioneered a probabilistic approach to this problem. If the number n of atoms in the unit cell of a crystal is large, these methods work very well. For smaller n, Hauptman suggested a direct algebraic approach, i.e., looking for explicit formulas expressing the magnitudes as rational functions in the observables. He found such formulas for n <= 3 and asked if similar formulas exist for other values of n. We show that the answer is ``yes" by rephrasing the problem as a question about multiplicative invariants of a particular finite group action. Our proofs are constructive, but the resulting algorithms are, for the most part, too slow to be of practical significance. For n <= 4, we develop a faster algorithm, using SAGBI bases. As a result, we obtain new explicit formulas for n = 4. We also consider several related problems and algorithms.
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Abstract: Let F be a field and i < j be integers between 1 and n. A map of Grassmannians f : Gr(i, F^n) --> Gr(j, F^n) is called nesting, if l is contained in f(l) for every l in Gr(i, F^n). We show that there are no continuous nesting maps over C and no algebraic nesting maps over any algebraically closed field F, except for a few obvious ones. As a corollary, we give a description of the algebraic (respectively, topological) subbundles of the tangent bundle to the projective space P^n over F (respectively, over C).
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Abstract: This is a note I wrote for the open problems chapter of the Proceedings of the Classical Invariant Theory Workshop held at Queens College in Kingston, Ontario in April, 2002. For an algebraic group G, defined over an algebraically closed field of characteristic zero, there is a natural partial order on the set of G-actions on algebraic varieties: X >= Y if there exists a dominant G-equivariant rational map (i.e., a compression) from X to Y. Alternatively, one can consider regular, rather than rational, compressions. In this note I propose to study this partial order in the case where G is a finite group. In particular, I am interested in describing the minimal elements (I call them incompressible G-varieties).
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Abstract: Let G be an algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed field of characteristic zero. It is well-known that X can be viewed as a G-torsor, representing a class [X] in H^1(K, G), where K is the field of G-invariant rational functions on X. We have previously shown that if X has a smooth H-fixed point for some non-toral diagonalizable subgroup of G then [X] is non-trivial. It is natural to ask if the converse is true, assuming G is connected and X is projective and smooth. In this note we show that the answer is ``no".
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Abstract: The subject matter of this paper falls somewhere between computational algebra, invariant theory, and the combinatorics of polyhedral cones in R^n. The main result is as follows. Let k be a field and G be a finite subgroup of GL_n(Z). We show that the ring of multiplicative invariants k[x_1, 1/x_1, \dots, x_n, 1/x_n]^G has a finite SAGBI basis if and only if G is generated by reflections.
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Abstract: This is a significantly revised and expanded version of the earlier preprint ``Lattices and parameter reduction in division algebras", written jointly with M. Lorenz (see below). Let A be a finite-dimensional division algebra containing a base field k in its center F. We say that A is defined over a subfield F_0 if there exists an F_0-algebra A_0 such that A = A_0 \otimes_{F_0} F. We ask when F_0 can be chosen to be rational over k or have low transcendence degree over k. We show that: (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n >= 5, then A is defined over a field F_0 of transcendence degree <= 1/2 (n-1)(n-2) over k. (iii) A is an algebra of degree 4 then A is Brauer equivalent to a tensor product of two symbol algebras. Consequently, M_2(A) can be defined over a field F_0 such that trdeg_k(F_0) <= 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F_0 of transcendence degree 4. (In (i), (iii) and (iv) we assume that the center of A contains certain roots of unity.) Our results in (iii) and (iv) complement a recent theorem of Rost, which asserts that a generic degree 4 algebra cannot be defined over a field F_0 with <= 4.
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Abstract: Let L/K be a G-Galois field extension and let G_2 be a Sylow subgroup of G. We show that if G_2 is not abelian then the trace form of L/K is hyperbolic, provided that K contains certain roots of unity.
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Abstract: We construct a birational invariant for certain algebraic group actions. We use this invariant to classify linear representations of finite abelian groups up to birational equivalence, thus answering, in a special case, a question of E. B. Vinberg and giving a family of counterexamples to a related conjecture of P. I. Katsylo. We also give a new proof of a theorem of M. Lorenz on birational equivalence of quantum tori (in a slightly expanded form) by applying our invariant in the setting of PGL_n-varieties.
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Abstract: We prove an equivariant version of Hironaka's theorem on elimination of points of indeterminacy. Our proof relies on canonical resolution of singularities.
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Abstract: Suppose E/F is a field extension. We ask whether or not there exists an element of E whose characteristic polynomial has one or more zero coefficients in specified positions. We show that the answer is frequently ``no''. We also prove similar results for division algebras and show that the universal division algebra of degree n does not have an element of trace 0 and norm 1.
Abstract: Let k be an algebraically closed field
of characteristic
0 and let D be a division algebra
whose center F contains k. We shall say that D
can be reduced to r parameters if we can write
D =
D0 tensor F0
F, where D0 is a division algebra, the center
F0 of D0 contains k and trdegk(F0)
= r.
We show that every division algebra of odd degree n
can be reduced to
<= 1/2(n-1)(n-2)
parameters. Moreover, every crossed product division algebra of degree
n >= 4 can be reduced to <= log2(n)
û-
1)n + 1 parameters.
Our proofs of these results rely on lattice-theoretic techniques.
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Abstract: In this short note, aimed primarily at advanced high school students, we give a new inductive proof of Newton's identities for symmetric polynomials.
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Abstract: Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We show that if H^1(K, G) = { 1 } for some finitely generated field extension K_0/k of transcendence degree >= 3 then H^1(L, G) = { 1 } for every field extension L/k.
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Abstract: We give a new algebro-geometric proof of the ``Key Lemma" of Parusinski.
Abstract: Let G be an algebraic group, X a generically free G-variety, and K=k(X)^G. A field extension L of K is called a splitting field of X if the image of the class of X under the natural map H^1(K, G) ---> H^1(L, G) is trivial. If L/K is a (finite) Galois extension then Gal(L/K) is called a splitting group of X. We prove a lower bound on the size of a splitting field of X in terms of fixed points of nontoral abelian subgroups of G. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.
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Abstract: Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X' with the following property: the stabilizer of every point of X' is isomorphic to a semidirect product of a unipotent group and a diagonalizable group. As an application of this and related results, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.
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Abstract: We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our group G is S_n, these objects are field extensions, if G = O_n, they are quadratic forms, if G = PGL_n, they are division algebras (all of degree n), if G = G_2, they are octonion algebras, if G = F_4, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, which relate the essential dimension of G to cohomological invariants of principal G-bundles.
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Abstract: A classical theorem of Hermite and Joubert asserts that any field extension of degree n = 5 or 6 is generated by an element whose minimal polynomial is of the form
t^n + c_1 t^{n-1} + ... + c_{n-1} t + c_n,
with c_1= c_3 = 0. We show that this theorem fails for n = 3^m or 3^m + 3^l with m > l >= 0 (and more generally, for n= p^m or p^m + p^l with m > l >= 0, if 3 is replaced by another prime p). We also prove similar results for division algebras and use them to study the structure of the universal division algebra UD(n).
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Abstract: We revisit the classical problem of simplifying polynomials by means of Tschirnhaus transformations. We consider Tschirnhaus transformations involving (i) no auxiliary radicals, (ii) arbitrary radicals, (iii) odd degree radicals, and (iv) odd degree radicals and the square root of the discriminant. We previously showed that by using substitutions of type (i) one cannot reduce the general polynomial of degree n to a form with less than [n/2] independent coefficients. In this paper we give a new proof of this result and also extend it to transformations of types (iii) and (iv). In the last section we present alternative proofs, based on the cohomological approach shown to us by J.-P. Serre.
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Abstract: Let V be a vector space and let {e_1, ... , e_r } be a basis of V. An algebra structure on V is given by r^3 structure constants c_{ij}^h, where e_i e_j = Sum_h c_{ij}^h e_h. We require this algebra structure to be associative with unit element e_1. This limits the sets of structure constants (c_{ij}^h) to a subvariety of k^{r^3}, which we denote by Alg_r. Base changes in V (leaving e_1 fixed) give rise to the natural transport of structure action on Alg_r; isomorphism classes of r-dimensonal algebras are in 1-1 correspondence with the orbits under this action.
In this paper we classify the smooth closed subvarieties of Alg_r which are invariant under the transport of structure action and study the singularities which may occur. In particular, we show that if r = n^2 then the closure of the locus corresponding to the matrix algebra M_n(k) is not smooth for n >= 3. This gives a negative answer to a question of Seshadri.
Abstract: Let A be a finite-dimensional central simple algebra and let k be a subfield of its center Z(A). We say that z_1, ... , z_m generate A as a central simple algebra over k if A = S^{-1}k[z_1, ..., z_m], where S is the intersection of k[z_1, ... , z_m] and Z(A)*. In particular, if A is a division algebra, this means that z_1, ..., z_m generate A as a division algebra in the usual sense. A theorem of Procesi gives a necessary condition for A to be generated by m elements as a central simple algebra over k. We show that this condition is also sufficient.
Abstract: Let f(x) = \sum a_i x^i be a monic polynomial of degree n whose coefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) is generating (for the symmetric group) if it can be obtained from f(x) by a non-degenerate Tschirnhaus transformation. We show that the minimal number d_k(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13-th problem.
Our approach to this question (and generalizations) is based on the idea of the ``essential dimension'' of a finite group G: the smallest possible dimension of an algebraic G-variety over k to which one can ``compress'' a faithful linear representation of G. We show that d_k(n) is just the essential dimension of the symmetric group S_n. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.