Preliminaries

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 22 February 2013

Preliminaries

Fix the following data and notation:

$$\begin{array}{cc}{𝔥}^{*}& \text{is a real vector space of dimension}\hspace{0.17em}n,\\ R& \text{is a reduced irreducible root system in}\hspace{0.17em}{𝔥}^{*},\\ R^{+}& \text{is a set of positive roots in}\hspace{0.17em}R,\\ W& \text{is the Weyl group of}\hspace{0.17em}R,\\ s_1,\dots ,s_n& \text{are the simple reflections in}\hspace{0.17em}W,\\ m_{ij}& \text{is the order of}\hspace{0.17em}{s}_i{s}_j\hspace{0.17em}\text{in}\hspace{0.17em}W,i\ne j,\\ R\left(w\right)=\{\alpha \in R^{+}\hspace{0.17em}\mid \hspace{0.17em}w\alpha \notin R^{+}\}& \text{is the inversion set of}\hspace{0.17em}w\in W,\\ \ell \left(w\right)=\text{Card}\left(R\left(w\right)\right)& \text{is the length of}\hspace{0.17em}w\in W,\\ \le & \text{is the Bruhat-Chevalley order on}\hspace{0.17em}W,\\ {\alpha }_1,\dots ,{\alpha }_n& \text{are the simple roots in}\hspace{0.17em}{R}^{+},\\ {\omega }_1,\dots ,{\omega }_n& \text{are the fundamental weights,}\\ P={\sum }_{i=1}^n\mathbb{Z}{\omega }_i& \text{is the weight lattice,}\\ P^{+}={\sum }_{i=1}^n{\mathbb{Z}}_{\ge 0}{\omega }_i& \text{is the set of dominant integral weights.}\end{array}$$

For a brief, easy, introduction to root systems with lots of pictures for visualization see [NRa2003]. By [Bou1981, VI Section 1 no. 6 Corollary 2 to Proposition 17], if $w=s_{i_1}\dots s_{i_p}$ is a reduced word for $w,$ then

$$\begin{array}{cc}R\left(w\right)=\{{\alpha }_{i_p},s_{i_p}{\alpha }_{i_{p-1}},\dots ,s_{i_p}\dots s_{i_2}{\alpha }_{i_1}\}\text{.}& \text{(1.1)}\end{array}$$

The affine nil-Hecke algebra is the algebra $\stackrel{\sim }{H}$ given by generators $T_1,\dots ,T_n$ and $X^{\lambda },$ $\lambda \in P,$ with relations

$$\begin{array}{cc}{T}_i^2=T_i,\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_i{T}_j{T}_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_j{T}_i{T}_j\dots },\hspace{0.17em}{X}^{\lambda }{X}^{\mu }=X^{\lambda +\mu },& \text{(1.2)}\end{array}$$

and

$$\begin{array}{cc}{X}^{\lambda }{T}_i=T_i{X}^{s_i\lambda }+\frac{x^{\lambda }-X^{s_i\lambda }}{1-X^{-{\alpha }_i}}\text{.}& \text{(1.3)}\end{array}$$

Let $T_w=T_{i_1}\dots T_{i_p}$ for a reduced word $w=s_{i_1}\dots s_{i_p}\text{.}$ Then

$$\begin{array}{cc}\{X^{\lambda }{T}_w\hspace{0.17em}\mid \hspace{0.17em}w\in W,\lambda \in P\}\text{and}\{T_w{X}^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}w\in W,\lambda \in P\}& \text{(1.4)}\end{array}$$

are bases of $\stackrel{\sim }{H}\text{.}$

Both the nil-Hecke algebra,

$$\begin{array}{cc}H=\mathbb{Z}\text{-span}\hspace{0.17em}\{T_w\hspace{0.17em}\mid \hspace{0.17em}w\in W\},\text{and}\mathbb{Z}\left[X\right]=\mathbb{Z}\text{-span}\hspace{0.17em}\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in P\}& \text{(1.5)}\end{array}$$

are subalgebras of $\stackrel{\sim }{H}\text{.}$ The action of $W$ on $\mathbb{Z}\left[X\right]$ is given by defining

$$\begin{array}{cc}w{X}^{\lambda }=X^{w\lambda },\text{for}\hspace{0.17em}w\in W,\lambda \in P,& \text{(1.6)}\end{array}$$

and extending linearly. The proof of the following theorem is given in [Ram2003, Theorem 1.13 and Theorem 1.17]. The first statement of the theorem is due to Bernstein, Zelevinsky, and Lusztig [Lus1983, 8.1] and the second statement is due to Steinberg [Ste1975] and is known as the Pittie–Steinberg theorem.

Theorem 1.7 Define

$$\begin{array}{cc}{\lambda }_w=w^{-1}\sum _{s_{i}w<w}{\omega }_i,\text{for}\hspace{0.17em}w\in W\text{.}& \text{(1.8)}\end{array}$$

The center of $\stackrel{\sim }{H}$ is $Z\left(\stackrel{\sim }{H}\right)=\mathbb{Z}{\left[X\right]}^W$ and each element $f\in \mathbb{Z}\left[X\right]$ has a unique expansion

$$\begin{array}{cc}f=\sum _{w\in W}{f}_w{X}^{-{\lambda }_w},\text{with}\hspace{0.17em}{f}_w\in \mathbb{Z}{\left[X\right]}^W\text{.}& \text{(1.9)}\end{array}$$

Let ${\epsilon }_i=1-T_i$ and let ${\epsilon }_w={\epsilon }_{i_1}\dots {\epsilon }_{i_p}$ for a reduced word $w=s_{i_1}\dots s_{i_p}\text{.}$ Then ${\epsilon }_w$ is well defined and independent of the reduced word for $w$ since

$$\begin{array}{cc}{\epsilon }_i^2={\epsilon }_i,\text{and}\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\epsilon }_i{\epsilon }_j{\epsilon }_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\epsilon }_j{\epsilon }_i{\epsilon }_j\dots }\text{.}& \text{(1.10)}\end{array}$$

The second equality is a consequence of the formulas

$$\begin{array}{cc}{\epsilon }_w=\sum _{v\le w}{(-1)}^{\ell \left(v\right)}{T}_v\text{and}{T}_w=\sum _{v\le w}{(-1)}^{\ell \left(v\right)}{\epsilon }_v& \text{(1.11)}\end{array}$$

which are straightforward to verify by induction on the length of $w\text{.}$

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and the Schubert calculus authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.

page history