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:
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
is a reduced word for then
The affine nil-Hecke algebra is the algebra given by generators
and
with relations
and
Let
for a reduced word
Then
are bases of
Both the nil-Hecke algebra,
are subalgebras of The action of on
is given by defining
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 PittieāSteinberg theorem.
Theorem 1.7 Define
The center of is
and each element has a unique expansion
Let and let
for a reduced word
Then is well defined and independent of the reduced word for since
The second equality is a consequence of the formulas
which are straightforward to verify by induction on the length of
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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