## Notes on Schubert Polynomials

Notes and References

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

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 ${\U0001d516}_{w}$ 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.

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