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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

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14M12 Determinantal varieties
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##### References:
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