Affine root systems of classical type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 26 January 2013

Affine root systems

The notation for affine root systems is WS = {sa  |  aS} and S=(S1,S2), where S1 = {aS  |  12aS} and S2 = {aS  |  2aS}.

Define O1 = {±εi+r  |  1in,r} O2 = {±2εi+2r  |  1in,r} O3 = {±εi+12(2r+1)  |  1in,r} O4 = {±2εi+2r+1  |  1in,r} O5- = {±(εi-εj)+r  |  1i<jn,r} O5+ = {±(εi+εj)+r  |  1i<jn,r} and O5=O5-O5+ so that the affine root systems of classical type are missing Note that εnO1,  2εnO2,  ε1+12δO3,  2ε1+δO4, and εi-εi+1O5,   for   i=1,...,n-1.

(Cn,Cn) = C-BCnII O4 O3 (Cn,BCn) = C-BCnI (BCn,Cn) = C-BCnIV O2 O3 O2 O1 Cn = C-Bn (Bn,Bn) = B-BCn BCn = C-BCnIII Cn O3 O2 O1 O4 O4 Bn Bn = B-Cn O1 O2 Dn O5+ An-1 O5-
Where the left notation is the notation in [M03, §1.3] and the right notation is that of [BT]. The Weyl groups of these are WS = { WCn, for types   Cn Cn BCn (BCn,Cn) (Cn,BCn) (Cn,Cn) , WBn, for types   Bn Bn (Bn,Bn) , WDn, for type   Dn } where WCn = W0𝔥 WBn = W0𝔥_ WDn = W0_𝔥_ with W0=G2,1,n, W0_=G2,2,n and 𝔥 = i=1n εi, and 𝔥_ = {λ1ε1++λnεn  |  λ1++λn =0 mod 2}, SAY SOMETHING ABOUT WS orbits on S. HOW DO THESE COMPARE TO KAC AND SAHI-ION???

When n=2 define O1 = {±εi+r  |  1i2,r} O2 = {±2εi+2r  |  1i2,r} O3 = {±εi+12(2r+1)  |  1i2,r} O4 = {±2εi+2r+1  |  1i2,r} O5- = {±(ε1-ε2)+r  |  r} O5+ = {±(ε1+ε2)+r  |  r} Then the "classical" affine root systems of rank 2 are

(C2,C2) O3 O4 O2 O1 (BC2,C2) (C2,BC2) O2 O4 O3 O4 O3 BC2 (C2,C2) C2 C2 O1 O2 O1 O1 O3 O2 O4 O4 A1×A1 O5+ A1 O5-
When n=1 define O1 = {±ε1+r  |  r} O2 = {±2ε1+2r  |  r} O3 = {±ε1+12(2r+1)  |  r} O4 = {±2ε1+2r+1  |  r} Then the "classical" affine root systems of rank 1 are
(C1,C1) O3 O4 (BC1,C1) (C1,BC1) O2 O2 BC1 O4 A1     A1 O1 O1O3

Notes and References

These notes are from a study of Macdonald polynomials of type C in 2008-2009.



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