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The Hafnian and a Commutative Analogue of the Grassmann Algebra
No full textA close relationship between the determinant, the pfaffian, and the Grassmann algebra is well-known. In this paper, a similar relation between the permanent, the hafnian, and a commutative analogue of the Grassmann algebra is described. Using the latter, some new properties of the hafnian are proved
Range-compatible homomorphisms over the field with two elements
No full textLet U and V be finite-dimensional vector spaces over a field K, and S be a linear subspace of the space L(U, V ) of all linear operators from U to V. A map F : S → V is called range-compatible when F(s) ∈ Im s for all s ∈ S. Previous work has classified all the range-compatible group homomorphisms provided that codimL(U,V )S ≤ 2 dim V − 3, except in the special case when K has only two elements and codimL(U,V )S = 2 dim V − 3. This article gives a thorough treatment of that special case. The results are partly based upon the recent classification of vector spaces of matrices with rank at most 2 over F2. As an application, the 2-dimensional non-reflexive operator spaces are classified over any field, and so do the affine subspaces of Mn,p(K) with lower-rank at least 2 and codimension 3
