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:

š”„* is a real vector space of dimensionā€‰n, R is a reduced irreducible root system inā€‰š”„*, R+ is a set of positive roots inā€‰R, W is the Weyl group ofā€‰R, s1,ā€¦,sn are the simple reflections inā€‰W, mij is the order ofā€‰sisjā€‰ inā€‰W,iā‰ j, R(w)= { Ī±āˆˆR+ā€‰āˆ£ ā€‰wĪ±āˆ‰R+ } is the inversion set ofā€‰wāˆˆW, ā„“(w)=Card (R(w)) is the length ofā€‰wāˆˆW, ā‰¤ is the Bruhat-Chevalley order onā€‰W, Ī±1,ā€¦,Ī±n are the simple roots inā€‰R+, Ļ‰1,ā€¦,Ļ‰n are the fundamental weights, P=āˆ‘i=1n ā„¤Ļ‰i is the weight lattice, P+= āˆ‘i=1n ā„¤ā‰„0Ļ‰i is the set of dominant integral weights.

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=si1ā€¦sip is a reduced word for w, then

R(w)= { Ī±ip, sip Ī±ip-1,ā€¦, sipā€¦si2 Ī±i1 } . (1.1)

The affine nil-Hecke algebra is the algebra Hāˆ¼ given by generators T1,ā€¦,Tn and XĪ», Ī»āˆˆP, with relations

Ti2=Ti, TiTjTiā€¦ āŸmijfactors = TjTiTjā€¦ āŸmijfactors ,ā€‰XĪ»XĪ¼= XĪ»+Ī¼, (1.2)

and

XĪ»Ti=Ti XsiĪ»+ xĪ»-XsiĪ» 1-X-Ī±i . (1.3)

Let Tw=Ti1ā€¦Tip for a reduced word w=si1ā€¦sip. Then

{ XĪ»Twā€‰āˆ£ā€‰ wāˆˆW,Ī»āˆˆP } and { TwXĪ»ā€‰āˆ£ā€‰ wāˆˆW,Ī»āˆˆP } (1.4)

are bases of Hāˆ¼.

Both the nil-Hecke algebra,

H=ā„¤-spanā€‰ {Twā€‰āˆ£ā€‰wāˆˆW}, andā„¤[X]=ā„¤ -spanā€‰ {XĪ»ā€‰āˆ£ā€‰Ī»āˆˆP} (1.5)

are subalgebras of Hāˆ¼. The action of W on ā„¤[X] is given by defining

wXĪ»=XwĪ», forā€‰wāˆˆW,Ī»āˆˆP, (1.6)

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 Pittieā€“Steinberg theorem.

Theorem 1.7 Define

Ī»w=w-1 āˆ‘siw<w Ļ‰i,for ā€‰wāˆˆW. (1.8)

The center of Hāˆ¼ is Z(Hāˆ¼)=ā„¤ [X]W and each element fāˆˆā„¤[X] has a unique expansion

f=āˆ‘wāˆˆWfw X-Ī»w, withā€‰fwāˆˆā„¤ [X]W. (1.9)

Let Īµi=1-Ti and let Īµw=Īµi1ā€¦Īµip for a reduced word w=si1ā€¦sip. Then Īµw is well defined and independent of the reduced word for w since

Īµi2=Īµi, and ĪµiĪµjĪµiā€¦ āŸmijfactors = ĪµjĪµiĪµjā€¦ āŸmijfactors . (1.10)

The second equality is a consequence of the formulas

Īµw=āˆ‘vā‰¤w (-1)ā„“(v) TvandTw= āˆ‘vā‰¤w (-1)ā„“(v) Īµv (1.11)

which are straightforward to verify by induction on the length of w.

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.

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