For those who want a linear representation of the week's news, below you can find WIRED's extensive Cambridge Analytica coverage—which dates back nearly two years.
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In this case \rho is a linear representation of G of degree 2.
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If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.
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While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a Lie group G, it does not imply that G has a faithful linear representation (which is not true in general), but rather that G always has a linear representation that is a local isomorphism with a linear group.
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PO is of basic interest in representation theory: a group homomorphism G → PGL is called a projective representation of G, just as a map G → GL is called a linear representation of G, and just as any linear representation can be reduced to a map G → O (by taking an invariant inner product), any projective representation can be reduced to a map G → PO. See projective linear group: representation theory for further discussion.
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Looking at the linear representation of these isomorphic algebras shows agreement in the fourth dimension when the negative sign is used; consider the sample product given above under linear representation. The University of Kansas has contributed to the development of bicomplex analysis. In 1953, Ph.D. student James D. Riley's thesis "Contributions to the theory of functions of a bicomplex variable" was published in the Tohoku Mathematical Journal (2nd Ser., 5:132–165).
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In mathematics, the Jacquet module is an module used in the study of automorphic representations. The Jacquet functor is the functor that sends a linear representation to its Jacquet module. They are both named after Hervé Jacquet.
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A projective representation of G can be pulled back to a linear representation of a central extension C of G. A group homomorphism G → PGL(V) from a group G to a projective linear group is called a projective representation of the group G, by analogy with a linear representation (a homomorphism G → GL(V)). These were studied by Issai Schur, who showed that projective representations of G can be classified in terms of linear representations of central extensions of G. This led to the Schur multiplier, which is used to address this question.
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A version of Fitting's lemma is often used in the representation theory of groups. This is in fact a special case of the version above, since every K-linear representation of a group G can be viewed as a module over the group algebra KG.
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Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).
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In mathematics, a linear representation ρ of a group G is a monomial representation if there is a finite-index subgroup H and a one-dimensional linear representation σ of H, such that ρ is equivalent to the induced representation :IndHGσ. Alternatively, one may define it as a representation whose image is in the monomial matrices. Here for example G and H may be finite groups, so that induced representation has a classical sense. The monomial representation is only a little more complicated than the permutation representation of G on the cosets of H. It is necessary only to keep track of scalars coming from σ applied to elements of H.
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In mathematics, a Molien series is a generating function attached to a linear representation ρ of a group G on a finite-dimensional vector space V. It counts the homogeneous polynomials of a given total degree d that are invariants for G. It is named for Theodor Molien.
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Kauders: Systematik der Drehstromwicklungen II, Elektrotechnik und Maschinenbau, 1934, vol. 8, pp. 85–92 the algebraic methods developed in the first paper were visualised by means of Tingley's diagram. The latter could be referred to as a linear representation of the now called star of slots (German: Nutenstern).
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2 One can extend the notion of tensor products to any finite number of representations. If V is a linear representation of a group G, then with the above linear action, the tensor algebra T(V) is an algebraic representation of G; i.e., each element of G acts as an algebra automorphism.
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If is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to . The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of , and the irreducible projective representations of , are essentially the same objects.
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Monomial matrices occur in representation theory in the context of monomial representations. A monomial representation of a group G is a linear representation ρ : G → GL(n, F) of G (here F is the defining field of the representation) such that the image ρ(G) is a subgroup of the group of monomial matrices.
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In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation.
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In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation.
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Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theorem states that L has a linear representation ρ over K, on a finite-dimensional vector space V, that is a faithful representation, making L isomorphic to a subalgebra of the endomorphisms of V.
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He left two conjectures, both known as Artin's conjecture. The first concerns Artin L-functions for a linear representation of a Galois group; and the second the frequency with which a given integer a is a primitive root modulo primes p, when a is fixed and p varies. These are unproven; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.
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An affine representation of a topological (Lie) group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) upon the Euclidean space En. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space: in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.
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In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties. Finite direct sums and products of rational representations are rational. A rational G module is a module that can be expressed as a sum (not necessarily direct) of rational representations.
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On the other hand, a hierarchical representation would have multiple levels of representation. A break in the link between lower level nodes does not render any part of the sequence inaccessible, since the control nodes (chunk nodes) at the higher level would still be able to facilitate access to the lower level nodes. Schematic of a hierarchical sequential structure with three levels. The lowest level could be a linear representation, while intermediate levels denote chunk nodes.
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Suppose Γ is an irreducible lattice in G. For a local field F and ρ a linear representation of the lattice Γ of the Lie group, into GLn (F), assume the image ρ(Γ) is not relatively compact (in the topology arising from F) and such that its closure in the Zariski topology is connected. Then F is the real numbers or the complex numbers, and there is a rational representation of G giving rise to ρ by restriction.
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A ring showing, conceptually, a circular buffer. This visually shows that the buffer has no real end and it can loop around the buffer. However, since memory is never physically created as a ring, a linear representation is generally used as is done below. In computer science, a circular buffer, circular queue, cyclic buffer or ring buffer is a data structure that uses a single, fixed-size buffer as if it were connected end-to-end.
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The quotient of the 24-dimensional linear representation of the permutation representation by its 1-dimensional fixed subspace gives a 23-dimensional representation, which is irreducible over any field of characteristic not 2 or 3, and gives the smallest faithful representation over such fields. Reducing the 24-dimensional representation mod 2 gives an action on F. This has invariant subspaces of dimension 1, 12 (the Golay code), and 23. The subquotients give two irreducible representations of dimension 11 over the field with 2 elements.
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In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:Lecture 1 of Section 4.3.3 : is the transpose of , that is, = for all . The dual representation is also known as the contragredient representation. If is a Lie algebra and is a representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:Lecture 8 of : = for all .
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Covering groups were introduced by Issai Schur to classify projective representations of groups. A (complex) linear representation of a group G is a group homomorphism G → GL(n,C) from the group G to a general linear group, while a projective representation is a homomorphism G → PGL(n,C) from G to a projective linear group. Projective representations of G correspond naturally to linear representations of the covering group of G. The projective representations of alternating and symmetric groups are the subject of the book .
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The double cover of the alternating group can be constructed via the spin representation that covers the usual linear representation of the alternating group. The double covers can be constructed as spin (respectively, pin) covers of faithful, irreducible, linear representations of An and Sn. These spin representations exist for all n, but are the covering groups only for n≥4 (n≠6,7 for An). For n≤3, Sn and An are their own Schur covers. the corresponding diagram of orthogonal and spin/pin groups.
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The advantages include: no backlash, high compactness and light weight, high gear ratios, reconfigurable ratios within a standard housing, good resolution and excellent repeatability (linear representation) when repositioning inertial loads, high torque capability, and coaxial input and output shafts. High gear reduction ratios are possible in a small volume (a ratio from 30:1 up to 320:1 is possible in the same space in which planetary gears typically only produce a 10:1 ratio). Disadvantages include a tendency for 'wind-up' (a torsional spring rate) in the low torque region.
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In mathematics, an invariant polynomial is a polynomial P that is invariant under a group \Gamma acting on a vector space V. Therefore, P is a \Gamma- invariant polynomial if :P(\gamma x) = P(x) for all \gamma \in \Gamma and x \in V. Cases of particular importance are for Γ a finite group (in the theory of Molien series, in particular), a compact group, a Lie group or algebraic group. For a basis-independent definition of 'polynomial' nothing is lost by referring to the symmetric powers of the given linear representation of Γ.
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In matroid theory, a mathematical discipline, the girth of a matroid is the size of its smallest circuit or dependent set. The cogirth of a matroid is the girth of its dual matroid. Matroid girth generalizes the notion of the shortest cycle in a graph, the edge connectivity of a graph, Hall sets in bipartite graphs, even sets in families of sets, and general position of point sets. It is hard to compute, but fixed-parameter tractable for linear matroids when parameterized both by the matroid rank and the field size of a linear representation.
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Many finite matroids may be represented by a matrix over a field K, in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type over a field F may also be represented as an algebraic matroid over F,Oxley (1992) p.220White (1987) p.24 by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals.
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More precisely, he obtained the result which is now known as Ado's Theorem: every finite- dimensional Lie algebra over a field of characteristic zero has a faithful finite-dimensional linear representation. After the defense of his doctoral thesis Igor Ado started to work at Kazan State University. From 1936 till 1942 he held the position of a professor at the Chair of Algebra. In 1942 Igor Ado moved to the Kazan State Chemical Technological Institute named after S.M. Kirov, now called Kazan National Research Technological University where he held the Chair of High Mathematics until his death.
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A module M over R[G] is then the same as a linear representation of G over the field R. There is no particular reason to limit R to be a field here. However, the classical results were obtained first when R is the complex number field and G is a finite group, so this case deserves close attention. It was shown that R[G] is a semisimple ring, under those conditions, with profound implications for the representations of finite groups. More generally, whenever the characteristic of the field R does not divide the order of the finite group G, then R[G] is semisimple (Maschke's theorem).
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In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex- analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis.
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The minimal degree of a faithful complex representation is 196,883, which is the product of the three largest prime divisors of the order of M. The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful complex representation. The smallest faithful permutation representation of the monster is on 24 · 37 · 53 · 74 · 11 · 132 · 29 · 41 · 59 · 71 (about 1020) points. The monster can be realized as a Galois group over the rational numbers , and as a Hurwitz group. The monster is unusual among simple groups in that there is no known easy way to represent its elements.
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Dynamic time warping is an approach that was historically used for speech recognition but has now largely been displaced by the more successful HMM-based approach. Dynamic time warping is an algorithm for measuring similarity between two sequences that may vary in time or speed. For instance, similarities in walking patterns would be detected, even if in one video the person was walking slowly and if in another he or she were walking more quickly, or even if there were accelerations and deceleration during the course of one observation. DTW has been applied to video, audio, and graphics – indeed, any data that can be turned into a linear representation can be analyzed with DTW.
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In the representation theory of finite groups, a vector space equipped with a group that acts by linear transformations of the space is called a linear representation of the group. A linear map that commutes with the action is called an intertwiner. That is, an intertwiner is just an equivariant linear map between two representations. Alternatively, an intertwiner for representations of a group over a field is the same thing as a module homomorphism of -modules, where is the group ring of G.. Under some conditions, if X and Y are both irreducible representations, then an intertwiner (other than the zero map) only exists if the two representations are equivalent (that is, are isomorphic as modules).
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In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly defined classes of lattices inside semisimple groups was the discovery of Grigory Margulis, who proved some fundamental results in this direction. There is more than one result that goes by the name of Margulis superrigidity. One simplified statement is this: take G to be a simply connected semisimple real algebraic group in GLn, such that the Lie group of its real points has real rank at least 2 and no compact factors.
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Structure of the NBD of ABC transporters with bound nucleotide (). Linear representation of protein sequence above shows the relative positions of the conserved amino acid motifs in the structure (colors match with 3D structure) The ABC domain consists of two domains, the catalytic core domain similar to RecA-like motor ATPases and a smaller, structurally diverse α-helical subdomain that is unique to ABC transporters. The larger domain typically consists of two β-sheets and six α helices, where the catalytic Walker A motif (GXXGXGKS/T where X is any amino acid) or P-loop and Walker B motif (ΦΦΦΦD, of which Φ is a hydrophobic residue) is situated. The helical domain consists of three or four helices and the ABC signature motif, also known as LSGGQ motif, linker peptide or C motif.
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Based on this principle, when an interviewer is conducting the interview he or she will receive better results using the first two retrieval rules if they are able to encourage the participant to recreate an overall environment similar to that of the event the participant had witnessed. For example, the interviewer could encourage the witness to recreate their original state (at the time of the crime) during the interview. Past research has demonstrated that memories that have been encoded during a high, emotionally aroused state may be accessible only if the same affect is reinstated during retrieval. The last two retrieval rules are based on the multi-component view of memory which maintains that memory trace is not a linear representation of the original event, but rather is a complex.
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Equivariant maps can be generalized to arbitrary categories in a straightforward manner. Every group G can be viewed as a category with a single object (morphisms in this category are just the elements of G). Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object. For example, a G-set is equivalent to a functor from G to the category of sets, Set, and a linear representation is equivalent to a functor to the category of vector spaces over a field, VectK. Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.
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In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1. For groups over finite fields, these representations were introduced by , first for the general linear groups, then for classical groups, and then for all Chevalley groups, with a construction that immediately generalized to the other groups of Lie type that were discovered soon after by Steinberg, Suzuki and Ree. Over a finite field of characteristic p, the Steinberg representation has degree equal to the largest power of p dividing the order of the group.
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Mumford defined a theta group associated to an invertible sheaf L on an abelian variety A. This is a group of self- automorphisms of L, and is a finite analogue of the Heisenberg group. The primary results are on the action of the theta group on the global sections of L. When L is very ample, the linear representation can be described, by means of the structure of the theta group. In fact the theta group is abstractly a simple type of nilpotent group, a central extension of a group of torsion points on A, and the extension is known (it is in effect given by the Weil pairing). There is a uniqueness result for irreducible linear representations of the theta group with given central character, or in other words an analogue of the Stone–von Neumann theorem.
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In the spin-foam formalism, the Barrett–Crane model, which was for a while the most promising state-sum model of 4D Lorentzian quantum gravity, was based on representations of the noncompact groups SO(3,1) or SL(2,C), so the spin foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks. These and other considerations, including difficulties interpreting what it would mean to apply a Lorentz transformation to a spin network state, led Lee Smolin and others to suggest that spin network states must break Lorentz invariance. Lee Smolin and Joao Magueijo then went on to study doubly special relativity, in which not only there is a constant velocity c but also a constant distance l. They showed that there are nonlinear representations of the Lorentz Lie algebra with these properties (the usual Lorentz group being obtained from a linear representation).
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For any linear representation W of G there is an associated vector bundle P\times^G W over M, and a principal connection induces a covariant derivative on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of P\times^G W over M is isomorphic to the space of G-equivariant W-valued functions on P. More generally, the space of k-forms with values in P\times^G W is identified with the space of G-equivariant and horizontal W-valued k-forms on P. If α is such a k-form, then its exterior derivative dα, although G-equivariant, is no longer horizontal. However, the combination dα+ωΛα is. This defines an exterior covariant derivative dω from P\times^G W-valued k-forms on M to P\times^G W-valued (k+1)-forms on M. In particular, when k=0, we obtain a covariant derivative on P\times^G W.
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The set of n×n generalized permutation matrices with entries in a field F forms a subgroup of the general linear group GL(n,F), in which the group of nonsingular diagonal matrices Δ(n, F) forms a normal subgroup. Indeed, the generalized permutation matrices are the normalizer of the diagonal matrices, meaning that the generalized permutation matrices are the largest subgroup of GL in which diagonal matrices are normal. The abstract group of generalized permutation matrices is the wreath product of F× and Sn. Concretely, this means that it is the semidirect product of Δ(n, F) by the symmetric group Sn: :Δ(n, F) ⋉ Sn, where Sn acts by permuting coordinates and the diagonal matrices Δ(n, F) are isomorphic to the n-fold product (F×)n. To be precise, the generalized permutation matrices are a (faithful) linear representation of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.
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A group G is said to be linear if there exists a field K, an integer d and an injective morphism from G to the general linear group GLd(K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K. Basic instances are groups which are defined as subgroups of a linear group, for example: #The group GLn(K) itself; #The special linear group SLn(K) (the subgroup of matrices with determinant 1); #The group of invertible upper (or lower) triangular matrices #If gi is a collection of elements in GLn(K) indexed by a set I, then the subgroup generated by the gi is a linear group. In the study of Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of complex numbers. (Some authors require that the group be represented as a closed subgroup of the GLn(C).) Books that follow this approach include Hall (2015) and Rossman (2002).
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In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V, without constant term, such that :F(v) ≠ 0\. The polynomial can be taken to be homogeneous, in other words an element of a symmetric power of the dual of V, and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p. When K has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of G implies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p was proved by W. J. , about a decade after the problem had been posed by David Mumford, in the introduction to the first edition of his book Geometric Invariant Theory.
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