--- title: Generalized Temperley–Lieb Trick url: https://www.emergentmind.com/topics/generalized-temperley-trick type: topic --- # Generalized Temperley–Lieb Trick The generalized Temperley–Lieb trick constitutes a comprehensive framework for transferring foundational polynomial constructs and recursive techniques of Kazhdan–Lusztig and R-polynomials from Hecke algebras to generalized Temperley–Lieb algebras of Coxeter groups. This paradigm enables the explicit construction, manipulation, and combinatorial analysis of polynomial families central to the representation theory of Coxeter groups and their related algebras, with particular emphasis on fully commutative elements and non-branching Coxeter graphs [1401.0611]. ## 1. Construction of Generalized Temperley–Lieb Algebras Given a fixed Coxeter system $(W,S)$ with associated length function $\ell$ and Bruhat order $\leq$, the Hecke algebra $\mathsf{H}(W)$ over $\mathbb{A} = \mathbb{Z}[q,q^{-1}]$ is equipped with generators $\{T_w : w \in W\}$ and an involution $\iota$ defined by $\iota(q) = q^{-1}$, $\iota(T_w) = (T_{w^{-1}})^{-1}$. The generalized Temperley–Lieb algebra $\mathrm{TL}(W)$ is constructed as the quotient of $\mathsf{H}(W)$ by the two-sided ideal $J$ generated by all $T_x$, where $x$ belongs to any rank–2 parabolic subgroup $\langle s_i, s_j \rangle$ having $m(s_i,s_j)<\infty$ and $s_is_j\neq s_js_i$: $$ \mathrm{TL}(W) := \mathsf{H}(W) / J. $$ Fully commutative elements, whose reduced expressions differ only by commutation, form the subset $W_c \subset W$. Graham's theorem establishes $\{t_w := T_w \bmod J : w \in W_c\}$ as an $\mathbb{A}$–basis for $\mathrm{TL}(W)$, termed the t–basis. For non-branching Coxeter graphs (excluding type $F_4$), the projection property asserts that the quotient homomorphism $o : \mathsf{H} \to \mathrm{TL}(W)$ satisfies $o(C_w) = 0$ for $w \notin W_c$ and $o(C_w) = c_w$ for $w \in W_c$, where $\{C_w\}_{w \in W}$ is the Kazhdan–Lusztig basis and $\{c_w\}_{w \in W_c}$ is the analogous IC–basis for $\mathrm{TL}(W)$. ## 2. Polynomial Families: R, D, a, and L Two polynomial families fulfill analogous roles in $\mathrm{TL}(W)$ to the R- and Kazhdan–Lusztig polynomials in $\mathsf{H}(W)$: - **R–polynomials ($R_{x,w}(q)$) in $\mathsf{H}(W)$**: Defined by the inversion formula $$ (T_{w^{-1}})^{-1} = \sum_{x \leq w} \varepsilon_x R_{x,w}(q) T_x, $$ with $\varepsilon_x = (-1)^{\ell(x)}$, $R_{w,w}(q)=1$, $R_{x,w}(q)=0$ for $x \not\leq w$. - **a–polynomials ($a_{x,w}(q)$) in $\mathrm{TL}(W)$**: Provide an analogous expansion $$ (t_{w^{-1}})^{-1} = q^{-\ell(w)} \sum_{x \in W_c, x \leq w} a_{x,w}(q)t_x, $$ where $a_{w,w}(q) = 1$. - **D–polynomials ($D_{x,w}(q)$)**: Serve as change–of–basis coefficients from arbitrary $w \in W$ to fully commutative $x$ $$ t_w = \sum_{x \in W_c, x \leq w} D_{x,w}(q) t_x, $$ with $D_{w,w}(q) = 1$. - **L–polynomials ($L_{x,w}(Q)$)**: When $\mathrm{TL}(W)$ admits an IC–basis, one writes $$ c_w = \sum_{x \in W_c, x \leq w} q^{\ell(x) - \ell(w)} L_{x,w}(q^{-1}) t_x. $$ Equivalently, with $Q = q^2$, $$ c_w = \sum_{x \leq w} Q^{(\ell(x)-\ell(w))/2} L_{x,w}(Q) t_x, $$ with $L_{w,w}=1$ and $L_{x,w}=0$ unless $x\leq w$. ## 3. Recursion Relations The families admit recursive formulations that extend classical relations: - **R–Polynomial Recursion (in $\mathsf{H}(W)$)**: For $s \in S$, $\ell(ws)<\ell(w)$, $$ R_{x,w}(q) = \begin{cases} R_{xs, ws}(q), & \ell(xs) < \ell(x) \\ q\, R_{xs, ws}(q) + (q-1) R_{x, ws}(q), & \ell(xs) > \ell(x) \end{cases} $$ - **D–Polynomial Recursion (Theorem 2.1)**: For $w \notin W_c$, $s \in S$ with $\ell(ws)<\ell(w)$, $ws \notin W_c$, and $x \in W_c, x \leq w$, $$ D_{x,w}(q) = \begin{cases} D_{x,ws}(q) + \displaystyle\sum_{y \in W_c, ys \notin W_c, ys > y} D_{x,ys}(q)D_{y,ws}(q), & \ell(xs)<\ell(x) \\ q D_{xs,ws}(q), & \ell(xs) > \ell(x), xs \notin W_c \\ D_{xs,ws}(q) + (q-1) D_{x,ws}(q), & \ell(xs) > \ell(x), xs \in W_c \end{cases} $$ - **a–Polynomial Recursion (Proposition 2.5)**: Under identical conditions, $$ a_{x,w}(q) = \begin{cases} a_{x,ws}(q) + \displaystyle\sum_{y \in W_c, ys \notin W_c, ys > y} D_{x,ys}(q) a_{y,ws}(q), & \ell(xs)<\ell(x) \\ -q a_{xs,ws}(q), & \ell(xs) > \ell(x), xs \notin W_c \\ a_{xs,ws}(q) + (1-q) a_{x,ws}(q), & \ell(xs) > \ell(x), xs \in W_c \end{cases} $$ Notably, if $xs \notin W_c$, $a_{x,w}(q) = -q a_{x,ws}(q)$ (Corollary 2.6). - **L–Polynomial Recursion**: - “Mixed–inversion” closed-form (Theorem 3.5) for non-branching graphs: $$ L_{x,w}(Q) = Q^{(\ell(x)-\ell(w))/2} \left[ P_{x,w}(q) + \sum_{x < y < w, y \in W_c} D_{x,y}(q) P_{y,w}(q) \right] $$ where $P_{x,w}(q)$ denotes the usual Kazhdan–Lusztig polynomial. - “Simple vanishing” and descent rules (Lemmas 3.6, Theorem 3.7): - If $s \in S$, $sw < w$, - $x < xs \notin W_c \implies L_{x,w}(Q) = 0$, - $x < xs \in W_c \implies L_{x,w}(Q) = q^{1-c} L_{sx,w}(Q)$, with $c=1$ if $sx < x$ and $c=0$ otherwise. ## 4. Closed–form Expressions - **Type $A$ closed-form for a–polynomials (Proposition 2.8)**: For $$ w = s_i s_{i+1}\cdots s_{i+k} s_{i-j}\cdots s_i\cdots s_{i+k-1}, \quad x = s_i s_{i+1}\cdots s_{i+k}, $$ $$ a_{x,w}(q) = (-q)^k (1-q)^j. $$ - The mixed-inversion formula for $L_{x,w}(Q)$ provides a closed-form in terms of $R$-, $D$-, and Kazhdan–Lusztig polynomials. ## 5. Combinatorial and Structural Properties All $D_{x,w}(q)$, $a_{x,w}(q)$, and $L_{x,w}(Q)$ lie in $\mathbb{Z}[q]$ or $\mathbb{Z}[Q]$. For non-branching Coxeter graphs, positivity results for $L$-coefficients parallel those in classical Kazhdan–Lusztig theory. For type $A$, the cardinality $|W_c(A_{n-1})|$ equals the $n$-th Catalan number, and numerous pattern-avoidance characterizations for $W_c$ are documented. The leading coefficient $M(x,w)$ of $L_{x,w}(Q)$ coincides with the leading coefficient $u(x,w)$ of $P_{x,w}(q)$, implying that “head” and “tail” phenomena of Kazhdan–Lusztig polynomials extend to their generalized counterparts. An orthogonality-type identity (Proposition 3.8) holds for the sum $$ F_w(Q) = \sum_{x