## Notes on Schubert PolynomialsNotes and References

Last update: 3 July 2013

## Notes and References

Chapter I. The notion of the diagram of a permutation $w$ is ascribed to J. Riguet in [LSc1982]. The code of $w$ is the Lehmer code, familiar to computer scientists. Vexillary permutations were introduced in [LSc1982] and enumerated in [LSc1985], though from a somewhat different point of view from that in the text.

Chapter II. Divided differences, in the context of an arbitrary root system, were introduced independently by Bernstein, Gelfand and Gelfand [BGG1973] and Demazure [Dem1974]. Both these papers establish (2.5), (2.10) and (2.13) in this more general context.

Chapter III. Multi-Schur functions were introduced, and the duality theorem (3.8) proved, by Lascoux [Las1974]. The proof of Sergeev's formula (3.12) is also due to Lascoux (private communication).

Chapter IV. Schubert polynomials, like divided differences, are defined in the context of an arbitrary root system in [BGG1973] and in [Dem1974]. What is special to the root systems of type A is the stability property (4.5), which ensures that the Schubert polynomial ${𝔖}_{w}$ is well-defined for all permutations $w\in {S}_{\infty }\text{.}$ Propositions (4.7), (4.8) and (4.9) are stated without proof in various places in [LSc1982,LSc1982-2,LSc1983,LSc1985,LSc1985-2,LSc1987,LSc1988] but as far as I am aware the only published proof of (4.9) is that of M. Wachs [Wac1985], which is different from the proof in the text, Proposition (4.15), appropriately modified, is valid for any root system, and in this more general form will be found in [BGG1973] and [Dem1974].

Chapter V. The scalar product (5.2) is introduced in [LSc1988]. The symmetry properties (5.23) of the coefficient matrices $\left({\alpha }_{uv}\right),$ $\left({\beta }_{uv}\right)$ are indicated in [LSc1987].

Chapter VI. Double Schubert polynomials were introduced in [Las1982]. For the interpolation forumla (6.8), see [LSc1985-2]. The generalization (6.20) of Sergeev's formula (3.12) is due to Lascoux (private communication).

Chapter VII. This chapter is mostly an amplification of [LSc1982-2]. Propositions (7.21)-(7.24) are due to Stanley [Sta1984].

## Notes and References

This is a typed excerpt of the book Notes on Schubert Polynomials by I. G. Macdonald.