We prove that the orbit closure of the determinant is not normal. A similar
result is obtained for the orbit closure of the permanent multiplied by a power
of a linear form.Comment: 28 page

Kumar, Shrawan

English

arXiv

Let v be a complex vector space of dimension m and let E := v ⊗ v∗ = End v. Consider det ∈ Q := Sm(E∗), where det is the function taking determinant of any X ∈ End v. Fix a basis {e1, . . . , em} of v and a positive integer n < mand consider the function p ∈ Q, defined by p(X) = xm−n 1,1 perm(Xo), Xo being the component of X in the right down n × n corner, where any element of End v is represented by a m×m-matrix X = (xi,j)1≤i,j,≤m in the basis {ei} and perm denotes the permanent. The group G = GL(E) canonically acts on Q. Let X (resp. Y) be the G-orbit closure of det (resp. p) inside Q. Then, X and Y are closed (affine) subvarieties of Q which are stable under the standard homothecy action of C∗ on Q. Thus, their affine coordinate rings C[X] and C[Y] are nonnegatively graded G-algebras over the complex numbers C. Clearly, EndE · det ⊂ X, where EndE acts on Q via: (g · q)(X) = q(gt · X), for g ∈ EndE, q ∈ Q and X ∈ E

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Geometry of orbits of permanents and determinants

https://cdr.lib.unc.edu/downloads/hq37vx55q

Geometry of orbits of permanents and determinants

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