Paths

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

Last update: 30 November 2012

Paths

Let $\lambda \in P$. The straight line path to $\lambda$ is the map

$$p_{\lambda }:\left[0,1\right]\to {𝔥}_{ℝ}^{*}\text{given by}{p}_{\lambda }\left(t\right)=\lambda t.$$

Let ${\ell }_1,{\ell }_2\in {ℝ}_{\ge 0}$. The concatention of maps $p_1:\left[0,{\ell }_1\right]\to {𝔥}_{ℝ}^{*}$ and $p_2:\left[0,{\ell }_2\right]\to {𝔥}_{ℝ}^{*}$ is the map $p_1\otimes p_2:[0,{\ell }_1+{\ell }_2]\to {𝔥}_{ℝ}^{*}$ given by $$(p_1\otimes p_2)\left(t\right)=\{\begin{array}{ll}{p}_1\left(t\right),& \text{for}t\in [0,{\ell }_1],\\ p_1\left({\ell }_1\right)+p_2(t-{\ell }_1),& \text{for}t\in [{\ell }_1,{\ell }_1+{\ell }_2],\end{array}$$ Let $r,\ell \in {ℝ}_{\ge 0}$. The r-stretch of a map $p:[0,\ell ]\to {𝔥}_{ℝ}^{*}$ is the map $rp:[0,r𝓁]\to {𝔥}_{ℝ}^{*}$ given by

$$\left(rp\right)\left(t\right)=r\cdot p(t/r).$$

The reverse of a map $p:[0,\ell ]\to {𝔥}_{ℝ}^{*}$ is the map $p^{*}:[0,\ell ]\to {𝔥}_{ℝ}^{*}$ given by

$$p^{*}\left(t\right)=p(\ell -t)-p\left(\ell \right).$$

The weight of a map $pcolon;[0,\ell ]\to {𝔥}_{ℝ}^{*}$ is the endpoint of p,

$$\mathrm{wt}\left(p\right)=p\left(\ell \right).$$

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 $p:[0,\ell ]\to {𝔥}_{ℝ}^{*}$ in $B_{\mathrm{univ}}$. Let $B$ be a set of paths (a subset of $B_{\mathrm{univ}}$). The character of $B$ is the element of $\mathbb{Z}\left[P\right]$ given by

$$\mathrm{char}\left(B\right)=\sum _{b\in B}{X}^{\mathrm{wt}\left(p\right)}.$$

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

$$\begin{array}{rc}{\stackrel{\sim }{e}}_i:B_{\mathrm{univ}}\to B_{\mathrm{univ}}\cup \left\{0\right\}& 1<i<n\\ {\stackrel{\sim }{f}}_i:B_{\mathrm{univ}}\to B_{\mathrm{univ}}\cup \left\{0\right\}& 1<i<n\end{array}$$

which are defined and constructed below, in Proposition 5.7 and Theorem 5.8. The crystal graph of B is the graph with

$$\text{vertices}B\text{and}\text{labeled edges}p\prime \stackrel{i}{\leftarrow }p\text{if}p\prime ={\stackrel{\sim }{f}}_{i}p.$$

i-strings

Let $B$ be a crystal. Let $p\in B$ and fix $i$, ($1\le i\le n$). The i-string of $p$ is the set of paths $S_i\left(p\right)$ generated from $p$ by applications of the operators ${\stackrel{\sim }{e}}_i$ and ${\stackrel{\sim }{f}}_i$.

The head of $S_i\left(p\right)$ is $h\in S_i\left(p\right)$ such that ${\stackrel{\sim }{e}}_{i}h=0$. The tail of $S_i\left(p\right)$ is $t\in S_i\left(p\right)$ such that ${\tilde{f}}_{i}t=0$.

The weights of the paths in $S_i\left(p\right)$ are $$\mathrm{wt}\left(t\right)=s_i\mathrm{wt}\left(h\right)=\mathrm{wt}\left(h\right)-⟨\mathrm{wt}\left(h\right),{\alpha }_i^{\vee }⟩{\alpha }_i,\dots ,\mathrm{wt}\left(h\right)-2{\alpha }_i,\mathrm{wt}\left(h\right)-{\alpha }_i,\mathrm{wt}\left(h\right),$$ and the crystal graph of $S_i\left(p\right)$ 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 $$d_i^{+}\left(p\right)=\left(\text{distance from}h\text{to}p\right)\text{and}{d}_i^{-}\left(p\right)=\left(\text{distance from}p\text{to}t\right),$$ so that ${{\stackrel{\sim }{e}}_i}^{d_i^{+}\left(p\right)}p=h$ and ${{\stackrel{\sim }{f}}_i}^{d_i^{-}\left(p\right)}p=t$.

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].

References

[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

page history