Generalized Temperley–Lieb Trick

• Generalized Temperley–Lieb trick is a framework that extends recursive polynomial methods from Hecke algebras to quotients defined by fully commutative elements in Coxeter groups.
• It provides explicit constructions of t–basis algebras and introduces polynomial families (R, a, D, and L) with well-defined recurrence relations and closed-form expressions.
• The approach reveals pattern-avoidance, positivity, and combinatorial properties, underpinning advanced applications in representation theory and algebraic combinatorics.

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 (Pesiri, 2014).

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

$$F_w(Q) = \sum_{x<w} \varepsilon_x Q^{\ell(x)} L_{x,w}(Q),$$

yielding $F_w(Q) = \delta_{e,w}$, analogous to the relation $\sum_x \varepsilon_x R_{x,w}(q) = 0$ for $w \neq e$ in $\mathsf{H}(W)$.

6. Canonical Example: Coxeter Graph $A_2$

In the Coxeter system $W = S_3$ with generators $S = \{s_1, s_2\}$, $m(s_1,s_2) = 3$, fully commutative elements are $\{e, s_1, s_2, s_1s_2, s_2s_1\}$; thus $\mathrm{TL}(A_2)$ has t–basis $\{t_e, t_1, t_2, t_{12}, t_{21}\}$.

• R–polynomials: $R_{e,s_1}(q) = R_{e,s_2}(q) = 1$, $R_{e,s_1s_2}(q) = q$, $R_{s_1,s_1s_2}(q) = 1$, $R_{s_2,s_1s_2}(q) = 0$.
• a–polynomials: $a_{e,s_1}(q) = 1$, $a_{s_1,s_1}(q) = 1$, $a_{e,s_1s_2}(q) = q$, $a_{s_1,s_1s_2}(q) = -1$.
• L–polynomials: $L_{e,s_1}(Q) = q$, $L_{s_1,s_1}(Q) = 1$; for $w = s_1s_2$, $L_{e,12}(Q) = q$, $L_{s_1,12}(Q) = q^2$, following $L_{s_1,12}(Q) = q L_{e,12}(Q)$.
• t–basis multiplication: $t_1 t_2 = t_{12}$, $t_2 t_1 = t_{21}$, $t_{12} t_1 = q t_{12} + (q-1) t_{21}$.

This explicit computation verifies recurrence relations, recovers classical Temperley–Lieb dimensions, and demonstrates appropriate t–basis multiplications.

7. Significance and Implications

The generalized Temperley–Lieb trick affords a parallelism between classical and generalized polynomial structures, enabling the transfer of recursive, combinatorial, and closed-form techniques from the Hecke algebra setting to a larger class of quotients influenced by Coxeter combinatorics. This framework underpins extensions of positivity, orthogonality, and pattern-avoidance phenomena, and enables explicit calculations in both classical and generalized contexts. A plausible implication is the broad applicability of these recursions and combinatorial insights beyond type $A$, subject to Coxeter graph conditions such as non-branchingness, thus illuminating new avenues in algebraic combinatorics and representation theory (Pesiri, 2014).