Paths and i-strings

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 22 January 2009

Paths

Let λ∈P. The straight line path to λ is the map

pλ:[0,1]→𝔥ℝ*   given by   pλ(t)=λt.

Let 𝓁1,𝓁2∈ℝ≥0 . The concatention of maps p1:[0,𝓁1]→𝔥ℝ* and p2:[0,𝓁2]→𝔥ℝ* is the map p1⊗p2:[0,𝓁1+𝓁2]→𝔥ℝ* given by

(p1⊗p2)(t)=p1(t), for  t∈[0,𝓁1], p1(𝓁1)+p2(t-𝓁1), for  t∈[𝓁1,𝓁1+𝓁2], Let r,𝓁∈ℝ≥0. The r-stretch of a map p:0𝓁→𝔥ℝ* is the map rp:0r𝓁 →𝔥ℝ* given by

rpt=r.pt/r .

The reverse of a map p:0𝓁→𝔥ℝ* is the map p*:0𝓁→𝔥ℝ* given by

p*t=p𝓁-t-p𝓁.

The weight of a map p:0𝓁→𝔥ℝ* is the endpoint of p

wtp=p𝓁.

Let

$$B_{\mathrm{univ}}$$

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:0𝓁→𝔥ℝ* in Buniv. Let B be a set of paths (a subset of Buniv). The character of B is the element of ℤP given by

charB=∑b∈BXwtp.

A crystal is the set of paths B that is closed under the action of the root operators

e∼i:Buniv→Buniv∪01<i<nf∼i:Buniv→Buniv∪01<i<n

which are defined and constructed below, in Proposition 5.7 and Theorem 5.8 ELSWHERE NOT INCLUDED. The crystal graph of B is the graph with

vertices   Bandlabeled edges   p'←ip  if  p'=f∼p.

i-strings

Let B be a crystal. Let p∈B and fix i,  1≤i≤n. The i-string of p is the set of paths Sip generated from p by applications of the operators e∼i and f∼i.

The head of Sip is h∈Sip such that e∼ih=0. The tail of Sip is t∈Sip such that f∼it=0.

The weights of the paths in Sip are

wtt=siwth=wth-wthαi∨αi,  …,wth-2αi,wth-αi,wth, and the crystal graph of Sip is

$\begin{array}{rcllllllllllll}|& \leftarrow & & d_i^{-}\left(p\right)& & \to & |& & & & & & & \\ t& \stackrel{i}{\leftarrow }& {\stackrel{\sim }{e}}_{i}t\stackrel{i}{\leftarrow }& \dots & \stackrel{i}{\leftarrow }{\stackrel{\sim }{f}}_{i}p& \stackrel{i}{\leftarrow }& p& \stackrel{i}{\leftarrow }& {\stackrel{\sim }{e}}_{i}p\stackrel{i}{\leftarrow }& \dots & \stackrel{i}{\leftarrow }& {\stackrel{\sim }{f}}_{i}h& \stackrel{i}{\leftarrow }& h\\ & & & & & & |& \leftarrow & & d_i^{+}\left(p\right)& & & \to & |\end{array}$

where di+p=distance fromhtopanddi-p=distance fromptot, so that e∼idi+pp=h and f∼idi-pp=t

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)