From: Boudewijn Moonen Subject: Re: map induced by exterior power Date: Tue, 08 Aug 2000 11:17:20 +0200 Newsgroups: sci.math To: Andreas Bieri Summary: [missing] Andreas Bieri wrote: > > Hi all, > > Let A:V->V denote a linear map and A#:V\wedge V -> V\wedge V its > induced map on the space of 'bivectors' of V, defined by > v\wedge w |-> Av \wedge Aw. > > My question is: > Given a linear map B:V\wedge V -> V\wedge V, when is B > induced by A (B=A#) ? Is there any explicit condition? > > Thanks in advance for any reply. > > Regards, > > Andreas Bieri > University of Bern, > NOSPAMbieria@math-stat.unibe.ch > (remove capital letters) To begin with, I cannot fully answer your question, I just can offer some outlines in which direction to look. Let us consider the following, slightly more general, situation. All vector spaces are over some field F and finite dimensional. If V is a vector space, k an integer between 0 and dim V, let A^k(V) denote the k-th exterior power of V. Given a linear map f: V --> W, this induces for each k a linear map A^k(f) : A^k(V) --> A^k(W) uniquely given by the property A^k(f)(v_1 wedge ... wedge v_k) := A^k(f)v_1 wedge ... wedge A^k(f)v_k for all v_1,...,v_k in V. This, in turn, defines a map P_k : Hom(V,W) --> Hom(A^k(V),A^k(W)) the image of which is an affine variety in Hom(A^k(V),A^k(W)), and you are asking for defining equations of this variety, at least for V = W and k = 2. Since your question is invariant under scalar multiplication, this amounts to giving equations of the projective variety G_k(V,W) := Proj(im(P_k)) in the projective space Proj(Hom(A^k(V),A^k(W))). By choosing bases, this can be translated into matrices. If M(m,n) denotes the space of m x n - matrices, the map P_k is P_k : M(m,n) --> M(b(m,k),b(n,k)) , where b(p,l) denotes the binomial p!/(l!(p-l)!), given as follows. Let the space M(m,n) have coordinates X_{ij}, so that the general matrix X in M(m,n) is X = (X_{ij}). Given sequences A = (i_1,... i_k), B = (j_1,...,j_k), ... of indices 1 <= i_1 < ... < i_k <= m, 1 <= j_1 < ... < j_k <= n,...., let X_{AB} denote the corresponding k X k - submatrix of X, i.e. the matrix (X_{i_p j_q}), and let [A|B] denote the determinant det X_{AB}, i.e. the corresponding k x k - minor of X. Then P_k is given by P_k(X) = ([A|B]) , the A and B running over all possible increasing sequences in lexicographic order. What your question, in the general setting, amounts to is to find all possible relations between the k x k - minors of a general m x n - matrix: geometrically, these define the projective variety G_k(V,W). Classically, some of these relations are known, which are quadratic in the [A|B]: these are the famous *Pluecker relations* (see [1], (4.4) Lemma on p. 41). In some cases, these are known to generate *all* relations, thus giving you the equations you want for these cases: A) k = m: In these cases, P_m is the famous *Pluecker embedding* and G_m(V,W) is the Grassmannian Grass_m(W) of m-dimensional subspaces of W embedded via the Pluecker embedding and known to be defined by the Pluecker relations (see [2]). The first nontrivial relation appears for k = m = 2, n = 4: [12|12][34|12] - [13|12][24|12] + [23|12][1412] = 0 defining the space of lines in projective 3-space as a quadric hypersurface in projective 5-space. B) m = n, k = m-1 (?): This should be in, or in-between, the lines of [3] (just a dark remembrance, no guarantee). So at least for m = n = 3 your question should be answered by the Pluecker relations. For general m = n, I do not know if they generate all possible relations; for this you should ask an expert in commutative algebra, and/or pose this question in sci.math.research. [1] W. BRUNS and U. VETTER, Determinantal Rings, SLNM 1327 [2] P. GRIFFITHS and J. HARRIS, Principles of Algebraic Geometry, J. Wiley 1978 [3] C. HUNEKE, Determinantal ideals of linear type, Arch. Math. 47 (1986), 324-329. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712