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

Last update: 30 November 2012


Let λP. The straight line path to λ is the map

pλ: [0,1] 𝔥* given by pλ(t) =λt.

Let 1,2 0. The concatention of maps p1:[0, 1] 𝔥* and p2: [0,2] 𝔥* is the map p1 p2: [0,1 +2] 𝔥* given by (p1 p2)(t) = { p1(t), for t[0, 1], p1(1) +p2(t -1), for t[1, 1+2] , Let r, 0. The r-stretch of a map p: [0,] 𝔥* is the map rp:[0, r𝓁] 𝔥* given by

(rp)(t) =rp(t/r) .
The reverse of a map p:[0,] 𝔥* is the map p*: [0,] 𝔥* given by
p*(t) =p(-t) -p().
The weight of a map pcolon;[0,] 𝔥* is the endpoint of p,
wt(p)=p() .

q p wt(p) wt(q)

2q pq

λ pλ (pq)*


be the set of maps generated by the straight line paths by operations of concatenation, stretching and reversing. A path is an element of p: [0,] 𝔥* in Buniv. Let B be a set of paths (a subset of Buniv). The character of B is the element of [P] given by
char(B) =bB Xwt(p).
A crystal is the set of paths B that is closed under the action of the root operators
ei :Buniv Buniv {0} 1<i<n fi :Buniv Buniv {0} 1<i<n
which are defined and constructed below, in Proposition 5.7 and Theorem 5.8. The crystal graph of B is the graph with
verticesB and labeled edges pip if p =fip.


Let B be a crystal. Let pB and fix i, (1in). The i-string of p is the set of paths Si(p) generated from p by applications of the operators ei and fi.

The head of Si(p) is hSi(p) such that eih =0. The tail of Si(p) is tSi(p) such that f˜it =0.

The weights of the paths in Si(p) are wt(t) =siwt(h) =wt(h)- wt(h), αi αi, , wt(h) -2αi, wt(h)-αi, wt(h), and the crystal graph of Si(p) is | di-(p) | t i eit i i fip i p i eip i i fih i h | di+(p) |

where di+(p) =(distance from hto p) and di-(p) =(distance from pto t), so that ei di+ (p) p=h and fi di- (p) p=t.

Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. This is a typed version of sections 5.2 and 5.3 of the paper [Ram2006].


[Ram2006] Alcove walks, Hecke algebras, Spherical functions, crystals and column strict tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (Special Issue: In honor of Robert MacPherson, Part 2 of 3) (2006) 963-1013. MR2282411

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