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Determinantal loci and the flag variety. (English) Zbl 0693.14021
The author asks for the connection between a recent generalization of the “second fundamental theorem” of invariant theory due to Abhyankar and the geometry of the flag variety \(FL(n)=\sqcup W(\tau) \), where the W(\(\tau)\) are the Bruhat cells of FL(n). The Zariski closure of a W(\(\tau)\) in FL(n) is called a Schubert variety X(\(\tau)\) in FL(n). The fundamental theorem claims the primality of the determinantal ideals \(I_ p(X)\subset k[X]\) where \(X=(X_{ij})\) is a matrix of indeterminates over a field k. In the generalization mentioned above one investigates ideals \(I_ p({\mathcal L})\), where \({\mathcal L}\) is a certain subset (ladder) of X. The corresponding affine variety is denoted by V(p,\(\mathcal L)\). In case \(\mathcal L\) is a rectangular ladder, \(V(p,\mathcal L)\) is a determinantal locus and therefore the corresponding X(\(\tau)\) is called a determinantal-type Schubert variety. In section 4 (see theorems 3, 5, 6) it is shown that in general \(V(p,\mathcal L)\) is an intersection of determinantal loci and consequently each Schubert variety \(X(\tau)\) in FL(n) is an intersection of determinantal-type Schubert varieties. There are \(\binom{n+1}3\) determinantal-type Schubert varieties of FL(n) of which \((n-1)\) are divisors in FL(n), and a determinantal-type Schubert variety has codimension \(\leq n^ 2/4\) in FL(n). These intersections are ideal-theoretic and the proofs are combinatorial. Therefore the main section of this paper is section 2 on bivectors and permutations which makes up almost all the paper.
Reviewer: M.Herrmann

14M15 Grassmannians, Schubert varieties, flag manifolds
14M12 Determinantal varieties
Full Text: DOI
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