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Positivity in the Grothendieck group of complex flag varieties. (English) Zbl 1052.14054

Let \(G\) be a complex semisimple simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X_:=G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, the classes \([{\mathcal O}_{X_w}]\) of structure sheaves of \(X_w\) form a \(\mathbb{Z}\)-basis of the \(K\)-group \(K(X)\) of \(X\). Write, under the product in \(K(X)\), for any \(u,\,v\in W/W_P\): \[ [{\mathcal O}_{X_u}]= \sum_{w\in W/W_P} c^w_{u,v}[{\mathcal O}_{X_w}], \] for some \(c^w_{u,v}\in \mathbb{Z}\).
Then, the main result of the paper under review asserts that \[ c^w_{u,v}(-1)^{\text{codim\,}X_u+ \text{codim\,}X_v+ \text{codim\,}X_w}\geq 0. \] This was conjectured by A. Buch [Acta Math. 189, 37–78 (2002; Zbl 1090.14015)] and proved by him for the Grassmannians.
The author, in fact, proves the following more general result asked by W. Graham [Duke Math. J. 102, 599–614 (2001; Zbl 1069.14055)]: Let \(Y\subset X\) be a closed subvariety with rational singularities. Express \[ [{\mathcal O}_Y]= \sum_{w\in W/W_P} c^w_Y[{\mathcal O}_{X_w}],\text{ for }c^w_Y\in \mathbb{Z}. \] Then, \((-1)^{\text{codim\,}X_w+ \text{codim\,}Y} c^w_Y\geq 0\).

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14M17 Homogeneous spaces and generalizations
13D15 Grothendieck groups, \(K\)-theory and commutative rings
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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References:

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