A Generalization of the Macdonald–You Formula Journal of Algebra204 (1998), 573–587 (doi:10.1006.jabr.1997.7342)
✍ Scribed by Piotr Pragacz
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 137 KB
- Volume
- 226
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
This note 1 concerns mainly restricting Schubert classes. An example of w x this operation was discussed in Proposition 8 of the original paper P . This example is revisited in the present note, and a combinatorial interpretation w x of the restriction coefficients is given based on Stembridge's results St .
w x Second, by comparing the formula appearing in the title of P with the results of Stembridge and some other combinatorial results, we deduce Ž some new identities in Propositions 2᎐6. They concern apart from the restrictions of Schubert classes to the cohomology of Lagrangian Grass-. mannians relations between Q-functions, Stembridge's coefficients, and various ''hook numbers.'' Moreover, we provide some examples illustrating w x P and the formulas given in the present note.
w x In this note, any unexplained notation or quotation stems from P . However, in order to make the notation maximally compatible with that w x Ž . used in St which is our principal reference here , we label strict partiw x tions by , and usually denotes an ordinary partition, contrary to P .
Before passing to the proper content of this note, we correct some w x points in P . Namely, due to some bugs in the computer system SCHUR w x w Ž .x Sch ; P, Example 3 b was miscalculated: the quadratic expression in Q-functions displayed there, written as a -ޚlinear combination of Q-functions, contains no negative summands. Consequently the sentence on p. w x Ž 585, lines 6᎐7 from the bottom, is to be withdrawn from P . These 1 This note was supported by KBN Grant 2P03A 05112.