Root operators

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

Last update: 28 February 2012

Root operators for the rank 1 case

Let $$B^{\otimes k}=\{b_1\otimes \cdots \otimes b_k | b_i\in B\},whereB=\{+1,-1,0\}.$$ Define $$\tilde{f}:B^{\otimes k}\to B^{\otimes k}\cup \left\{0\right\}and\tilde{e}:B^{\otimes k}\to B^{\otimes k}\cup \left\{0\right\}$$ as follows. Let $b=b_1\otimes \cdots \otimes b_k\in B^{\otimes k}.$ Ignoring 0s successively pair adjacent unpaired $(+1,-1)$ pairs to obtain a sequence of unpaired $-1s$ and $+1s$ $$-1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1$$ (after pairing and ignoring 0s). Then $$\begin{array}{ll}\begin{array}{rcl}\tilde{f}b& =& same\; as\; b \; except\; the\; leftmost\; unpaired\; +1 \; is\; changed\; to\; -1,\\ \tilde{e}b& =& same\; as\; b \; except\; the\; rightmost\; unpaired\; -1 \; is\; changed\; to\; +1.\end{array}& (Ro\; 30)\end{array}$$ $\begin{array}{l}If\; there\; is\; no\; unpaired\; +1 \; after\; pairing\; then\; \tilde{f}b=0.\\ If\; there\; is\; no\; unpaired\; -1 \; after\; pairing\; then\; \tilde{e}b=0.\end{array}$

These operators coincide with the operators used in the type A case by Lascoux and Schützenberger [LS] (see the nice exposition in [Ki]). The $(+1,-1)$ pairing procedure is equivalent to the process of taking the "outer edge" of the path (Psf 19)-(Psf 20). In the context of Section 4 this is natural since only the outer edge of the path contributes nontrivially to the image of the path in the affine Hecke algebra.

Let ${𝔥}_{ℝ}^{*}=ℝ.$ By identifying $+1, -1, 0$ with the straight line paths $$\begin{array}{rcrcrc}& p1 & & p-1 & & p0 \\ p_1:& [0,1]\to {𝔥}_{ℝ}^{*}& p_{-1}:& [0,1]\to {𝔥}_{ℝ}^{*}& p_0:& [0,1]\to {𝔥}_{ℝ}^{*}\\ & t\mapsto t,& & t\mapsto -t,& & t\mapsto 0,\end{array}$$ respectively, the set $B^{\otimes k}$ is viewed as a set of maps $p:[0,k]\to {𝔥}_{ℝ}^{*}.$ Let $B^{\otimes 0}=\left\{\phi \right\}$ with $\tilde{f}\phi =0$ and $\tilde{e}\phi =0.$ Then $$\begin{array}{ll}{T}_{\mathbb{Z}}\left(B\right)=\underset{k\in {\mathbb{Z}}_{\ge 0}}{⨆}{B}^{\otimes k}& (Ro\; 31)\end{array}$$

Note that since $⟨p_i\left(t\right),{\alpha }_i^{\vee }⟩=-1, p$ is not the head of its $i-$string and $s_i\circ p$ is well defined. If $q=s_i\circ p$ then the first time $t$ that $q$ leaves the cone $C-\rho$ and $p\left(t\right)=q\left(t\right).$ Thus $s_i\circ q=p$ and $s_i\circ (s_i\circ p)=p.$ Since $$s_{\mathrm{wt}(s_i\circ p)}=s_{s_i\circ \mathrm{wt}\left(p\right)}=-s_{\mathrm{wt}\left(p\right)}$$ the terms $s_{\mathrm{wt}(s_i\circ p)}$ and $s_{\mathrm{wt}\left(p\right)}$ cancel in the sum in (5.29). Thus $$\mathrm{char}\left(B\right)=\sum _{\genfrac{}{}{0ex}{}{p\in B}{p\subseteq C-\rho }}{s}_{\mathrm{wt}\left(p\right)}.$$

Recall the notation for Weyl characters, tensor products multiplicities, restriction multiplicities and paths from (5.5), (5.11), (5.17) and (5.22). For each $\lambda \in P^{+}$ fix a highest weight path $p_{\lambda }^{+}$ with endpoint $\lambda$ and let $$B\left(\lambda \right) \; be\; the\; crystal\; generated\; by\; p_{\lambda }^{+}.$$ Let $\lambda , \mu , \nu \in P^{+}$ and let $J\subseteq \{1,2,...,n\}.$ Then $$s_{\lambda }=\sum _{p\in B\left(\lambda \right)}{X}^{\mathrm{wt}\left(p\right)},s_{\mu }{s}_{\nu }=\sum _{\genfrac{}{}{0ex}{}{q\in B\left(\nu \right)}{p_{\mu }^{+}\otimes q\subseteq C-\rho }}{s}_{\mu +\mathrm{wt}\left(q\right)},and{s}_{\lambda }=\sum _{\genfrac{}{}{0ex}{}{p\in B\left(\lambda \right)}{p\subseteq C_J-{\rho }_J}}{s}_{\mathrm{wt}\left(p\right)}^J.$$

		Proof.
	The path $p_{\lambda }^{+}$ is the unique highest weight path in $B\left(\lambda \right).$ Thus, by Theorem 5.5, $\mathrm{char}\left(B\left(\lambda \right)\right)=s_{\lambda }.$ By the "Leibnitz formula" for the root operators in Theorem 2.1c the set $$B\left(\mu \right)\otimes B\left(\nu \right)=\{p\otimes q | p\in B\left(\mu \right), q\in B\left(\nu \right)\}$$ is a crystal. Since $\mathrm{wt}(p\otimes q)=\mathrm{wt}\left(p\right)+\mathrm{wt}\left(q\right),$ $$\begin{array}{rcl}{s}_{\mu }{s}_{\nu }& =& \mathrm{char}\left(B\left(\mu \right)\right)\mathrm{char}\left(B\left(\nu \right)\right)\\ & =& \mathrm{char}\left((B\left(\mu \right)\otimes B\left(\nu \right))\right)\\ & =& \sum _{\genfrac{}{}{0ex}{}{p\otimes q\in B\left(\mu \right)\otimes B\left(\nu \right)}{p\otimes q\subseteq C-\rho }}{s}_{\mathrm{wt}\left(p\right)+\mathrm{wt}\left(q\right)}\\ & =& \sum _{\genfrac{}{}{0ex}{}{q\in B\left(\nu \right)}{p_{\mu }^{+}\otimes q\subseteq C-\rho }}{s}_{\mu +\mathrm{wt}\left(q\right)},\end{array}$$ is a set of paths closed under products, reverses and $r-$stretches $(r\in {\mathbb{Z}}_{\ge 0})$ and the action of $\tilde{e}$ and $\tilde{f}.$ For $p\in B$ let $$\begin{array}{ll}\begin{array}{rcl}{d}^{+}\left(p\right)& =& (number\; of\; unpaired\; +1s\; after\; pairing),\\ d^{-}\left(p\right)& =& (number\; of\; unpaired\; -1s\; after\; pairing).\end{array}& (Ro\; 32)\end{array}$$ See the picture in [Ro 33]. These are the nonnegative integers such that $${\tilde{f}}^{d^{+}\left(p\right)}p\ne 0and{\tilde{f}}^{d^{+}\left(p\right)+1}p=0,and{\tilde{e}}^{d^{-}\left(p\right)}p\ne 0and{\tilde{e}}^{d^{-}\left(p\right)+1}p=0.$$ $\square$

Use notations for $T_{\mathbb{Z}}\left(B\right)$ as in [Ro 30]-[Ro 32].

1. $\begin{array}{l}\\ \\ \\ \\ \\ Ifp\in T_{\mathbb{Z}}\left(B\right)and\tilde{f}p\ne 0then\tilde{e}\tilde{f}p=p.\\ Ifp\in T_{\mathbb{Z}}\left(B\right)and\tilde{e}p\ne 0then\tilde{f}\tilde{e}p=p.\end{array}$
2. If $p\in T_{\mathbb{Z}}\left(B\right)$ and $r\in {\mathbb{Z}}_{\ge 0}$ then $${\tilde{f}}^r\left(rp\right)=r\left(\tilde{f}p\right)and{\tilde{e}}^r\left(rp\right)=r\left(\tilde{e}p\right).$$
3. If $p,q\in T_{\mathbb{Z}}\left(B\right)$ then $$\tilde{f}(p\otimes q)=\{\begin{array}{ll}\tilde{f}p\otimes q,& if\; d^{+}\left(p\right)>d^{-}\left(q\right),\\ p\otimes \tilde{f}q,& if\; d^{+}\left(p\right)\le d^{-}\left(q\right),\end{array}\phantom{\}}$$ and $$\tilde{e}(p\otimes q)=\{\begin{array}{ll}\tilde{e}p\otimes q,& if\; d^{+}\left(p\right)\ge d^{-}\left(q\right),\\ p\otimes \tilde{e}q,& if\; d^{+}\left(p\right)<d^{-}\left(q\right).\end{array}\phantom{\}}$$
4. If $p\in T_{\mathbb{Z}}\left(B\right)$ then $$\tilde{f}\left(p^{*}\right)={\left(\tilde{e}p\right)}^{*}and\tilde{e}\left(p^{*}\right)={\left(\tilde{f}p\right)}^{*}.$$

		Proof.
	(a), (b) and (d) are direct consequences of the definition of the operators $\tilde{e}$ and $\tilde{f}$ and the definitions of $r-$stretching and reversing. (c) View $p$ and $q$ as paths. After pairing, $p$ and $q$ have the form $$\begin{array}{ll}p=\begin{array}{l}\\ \\ d+(p) d-(p) \end{array}andq=\begin{array}{l}\\ \\ d+(p) d-(p) \end{array}& (Ro\; 33)\end{array}$$ where the left traveling portion of the path corresponds to $-1s$ and the right traveling portion of the path corresponds to $+1s.$ Then $${\tilde{f}}_i(p_1\otimes p_2)=\{\begin{array}{ll}\tilde{f}p\otimes q,& if\; p\otimes q=\begin{array}{c}\\ \\ \end{array}, \; i.e.\; d^{+}\left(p\right)>d^{-}\left(q\right),\\ p\otimes \tilde{f}q,& if\; p\otimes q=\begin{array}{c}\\ \\ \end{array}, \; i.e.\; d^{+}\left(p\right)\le d^{-}\left(q\right),\end{array}$$ since in the first case, the leftmost unpaired $+1$ is from $p$ and, in the second case, it is from $q.$ $\square$

Use property (b) in Proposition 1.2 to extend the operators $\tilde{e}$ and $\tilde{f}$ to operators on $T_{\mathbb{Q}}\left(B\right),$ the set of maps $p:[0,l]\to ℝ$ generated by $B$ under the operations of concatenation, reversing and $r-$stretching $(r\in {\mathbb{Q}}_{\ge 0}).$ Then, by completion, the operators $\tilde{e}$ and $\tilde{f}$ extend to operators on $$\begin{array}{ll}\begin{array}{ll}{T}_{ℝ}\left(B\right),& the\; set\; of\; maps\; $ p:[0,l]\to ℝ$ \; generated\; by\; B \; by\; operations\; of\\ & concatenation,\; reversing\; and\; $ r-$stretching\; (r\in {ℝ}_{\ge 0}).\end{array}& (Ro\; 34)\end{array}$$ A rank 1 path is an element of $T_{ℝ}\left(B\right).$

The root operators in the general case

Recall from (5.23) that $$\begin{array}{rc}{B}_{\mathrm{univ}}& is\; the\; set\; of\; maps\; generated\; by\; the\; straight\; line\; paths\\ & by\; operations\; of\; concatenation,\; reversing\; and\; stretching,\end{array}$$ and a path is an element $p:[0,l]\to {𝔥}_{ℝ}^{*}$ in $B_{\mathrm{univ}}$ (see (5.23)).

Let $p:[0,l]\to ℝ$ be a path and let $\alpha \in R^{+}$ be a positive root. The map $$\begin{array}{ll}{p}_{\alpha }:[0,l]\to ℝgiven\; by{p}_{\alpha }\left(t\right)=⟨p\left(t\right),{\alpha }^{\vee }⟩& (Ro\; 35)\end{array}$$ is a rank 1 path (an element of $T_{ℝ}\left(B\right)$). The rank 1 path $p_{\alpha }$ is the projection of $p$ onto the line perpendicular to the hyperplane $H_{\alpha }.$ Define operators $$\begin{array}{ll}{\tilde{e}}_{\alpha }:B_{\mathrm{univ}}\to B_{\mathrm{univ}}\cup \left\{0\right\}and{\tilde{f}}_{\alpha }:B_{\mathrm{univ}}\to B_{\mathrm{univ}}\cup \left\{0\right\}& (Ro\; 36)\end{array}$$ by $$\begin{array}{ll}{\tilde{e}}_{\alpha }p=p+\frac{1}{2}(\tilde{e}{p}_{\alpha }-p_{\alpha })\alpha and{\tilde{f}}_{\alpha }p=p-\frac{1}{2}(p_{\alpha }-\tilde{f}{p}_{\alpha })\alpha ,& (Ro\; 37)\end{array}$$ and set $$\begin{array}{ll}{\tilde{e}}_i={\tilde{e}}_{{\alpha }_i}and{\tilde{f}}_i={\tilde{f}}_{{\alpha }_i},for\; 1\le i\le n.& (Ro\; 38)\end{array}$$ The operators ${\tilde{e}}_i$ and ${\tilde{f}}_i$ are designed so that after projection onto the line perpendicular to $H_{{\alpha }_i}$ they are the operators $\tilde{e}$ and $\tilde{f}.$

The dark parts of that path $p$ are reflected (in a mirror parallel to $H_{{\alpha }_i}$) to form the path ${\tilde{f}}_{i}p.$ The left dotted line is the affine hyperplane parallel to $H_{{\alpha }_i}$ which intersects the path $p$ at its leftmost (most negative) point (relative to $H_{{\alpha }_i}$) and the distance between the dotted lines is exactly the distance between lines of lattice points in $P$ parallel to $H_{{\alpha }_i}.$

The following theorem is a consequence of Proposition 1.2 and the definition in [Ro 34]. The uniqueness of the operators ${\tilde{f}}_i$ and ${\tilde{e}}_i$ is forced by the properties (b), (c) and (d) in Theorem 2.1.

The operators ${\tilde{e}}_i$ and ${\tilde{f}}_i$ defined in (Ro 38) are the unique operators such that

1. $\begin{array}{l}\\ \\ \\ \\ \\ If\; p\in B_{\mathrm{univ}} \; and\; {\tilde{f}}_{i}p\ne 0 \; then\; {\tilde{e}}_i{\tilde{f}}_{i}p=p.\\ If\; p\in B_{\mathrm{univ}} \; and\; {\tilde{e}}_{i}p\ne 0 \; then\; {\tilde{f}}_i{\tilde{e}}_{i}p=p.\end{array}$
2. If $\lambda \in P$ and $⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{>0}$ then $${\tilde{f}}_i^{⟨\lambda ,{\alpha }_i^{\vee }⟩}{p}_{\lambda }=p_{s_i\lambda }.$$
3. If $p,q\in B_{\mathrm{univ}}$ then $${\tilde{f}}_i(p\otimes q)=\{\begin{array}{ll}{\tilde{f}}_{i}p\otimes q,& if\; d_i^{+}\left(p\right)>d_i^{-}\left(q\right),\\ p\otimes {\tilde{f}}_{i}q,& if\; d_i^{+}\left(p\right)\le d_i^{-}\left(q\right),\end{array}\phantom{\}}and$$ $${\tilde{e}}_i(p\otimes q)=\{\begin{array}{ll}{\tilde{e}}_{i}p\otimes q,& if\; d_i^{+}\left(p\right)\ge d_i^{-}\left(q\right),\\ p\otimes {\tilde{e}}_{i}q,& if\; d_i^{+}\left(p\right)<d_i^{-}\left(q\right),\end{array}\phantom{\}}\phantom{and}$$ where $d_i^{\pm }\left(p\right)=d^{\pm }\left(p_{{\alpha }_i}\right)$ with $d^{\pm }$ as in (Ro 33) and $p_{{\alpha }_i}$ as in (Ro 35).
4. If $p\in B_{\mathrm{univ}}$ and $r\in {\mathbb{Z}}_{\ge 0}$ then $${\tilde{f}}_i^r\left(rp\right)=r\left({\tilde{f}}_{i}p\right)and{\tilde{e}}_i^r\left(rp\right)=r\left({\tilde{e}}_{i}p\right).$$
5. If $p\in B_{\mathrm{univ}}$ then $${\tilde{f}}_i{p}^{*}={\left({\tilde{e}}_{i}p\right)}^{*}and{\tilde{e}}_i{p}^{*}={\left({\tilde{f}}_{i}p\right)}^{*}.$$
6. $\begin{array}{l}\\ \\ \\ \\ \\ If\; p\in B_{\mathrm{univ}} \; and\; {\tilde{f}}_{i}p\ne 0 \; then\; \mathrm{wt}\left({\tilde{f}}_{i}p\right)=\mathrm{wt}\left(p\right)-{\alpha }_i.\\ If\; p\in B_{\mathrm{univ}} \; and\; {\tilde{e}}_{i}p\ne 0 \; then\; \mathrm{wt}\left({\tilde{e}}_{i}p\right)=\mathrm{wt}\left(p\right)+{\alpha }_i.\end{array}$

Notes and References

The above notes are taken from section 5 of the paper

[Ram] A. Ram, Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (2006), 963-1013.

specifically sections 5.5 and 5.6.

References

[BD] Y. Billig and M. Dyer, Decompositions of Bruhat type for the Kac-Moody groups, Nova J. Algebra Geom. 3 no. 1 (1994), 11-31.

[Br] M. Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 no. 1 (2002), 137-159.

[GL] S. Gaussent and P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 no. 1 (2005), 35-88.

[GR] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 no. 8 (2004), 1263-1283.

[Ha] T. Haines, Structure constants for Hecke and representation rings, Int. Math. Res. Not. 39 (2003), 2103-2119.

[IM] N. Iwahori and H. Hatsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $𝔭-$adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5-48.

[Ki] K. Killpatrick, A Combinatorial Proof of a Recursion for the q-Kostka Polynomials, J. Comb. Th. Ser. A 92 (2000), 29-53.

[KM] M. Kapovich and J.J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation theory, arXiv: math.RT/0411182.

[LS] A. Lascoux and M.P. Schützenberger, Le monoïde plaxique, Quad. Ricerce Sci. 109 (1981), 129-156.

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