Geometry of G/P. V
The paper under review is a continuation of the authors' earlier work on the socalled standard monomial theory for a semi-simple algebraic group G [Part I: C. P. Ramanujam.-A tribute. Collect. Publ. of C. P. Ramanujam and Pap. in his memory, Tata Inst. fundam. Res., Stud. Math. 8, 207-239 (1978) by \textit{C. S. Seshadri}; part II: Proc. Indian Acad. Sci., Sect. A 87, No.2, 1-54 (1978) by \textit{V. Lakshmibai} and \textit{C. S. Seshadri}, part III: ibid., Sect. A, Part III 88, No.2, 93-177 (1979) by \textit{V. Lakshmibai, C. Musili} and \textit{C. S. Seshadri}; part IV: ibid., Sect. A, Part III 88, No.4, 279-362 (1979), by the same three authors, see Zbl 0447.14010, Zbl 0447.14011, Zbl 0447.14012, Zbl 0447.14013]. See also the following review (Zbl 0618.14027) for subsequent papers. In part V a self-contained exposition of the main results of standard monomial theory is given. The case of Schubert varieties in G/P, where P is a maximal parabolic subgroup of ''classical type'' is treated. Although the method of proof is similar to the one presented in the earlier versions, an important modification has been made which makes it possible to avoid the use of Demazure's character formula. This formula played an essential role in the earlier work, and this caused trouble because a serious gap was found in Demazure's proof (this is no longer a problem since alternative proofs of the formula have been obtained by the second author, the reviewer and others). The starting point of the standard monomial theory is the socalled first basis theorem which states that there exists a basis (over \({\mathbb{Z}})\) indexed by admissible pairs for the global sections of the fundamental line bundle on G/P.
scientific article; zbMATH DE number 4002268
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