2.1. Coxeter groups

Let $(W, S), S = \{s_1, \ldots , s_n\},$ be a Coxeter system. Thus, $W$ is a group with a presentation $(s_0, \ldots , s_n | (s_is_j)^{m_{ij}} = e)$ , where $m_{ij} \in \mathbb{Z}^+ \cup \{\infty \}$ , with $m_{ii} = 1, m_{ji} = m_{ij} \geq 2$ if $i \neq j$ , and $m_{ij} = \infty$ means that there is no relation included for the pair $s_i, s_j$ . Such a presentation is read off the Coxeter diagram of $(W,S)$ in a standard way: such a diagram is a non-oriented graph with $S$ as a set of vertices, and $m_{ij} - 2$ edges between the vertices $s_i$ and $s_j$ . Let $l \, :\, W \to \mathbb{Z}_{\geq 0}$ be the Bruhat length function with respect to $S$ .

We will be primarily interested in Coxeter groups that are Weyl groups of classical types $A, B = C, D$ . To fix the numbering of the generators, their Coxeter diagrams are presented, with a vertex corresponding to $s_i \in S$ marked by $i$ . The Coxeter diagram of type $A_n$ is

of type $B_n = C_n$ is

and of type $D_n$ is

When $n$ is small, these are interpreted in a standard way as follows: the Coxeter system $A_1 = B_1 = D_1$ has a single vertex, $D_2 = A_1 \times A_1$ , $D_3 = A_3$ . We also adopt the convention $A_0 = B_0 = D_0$ corresponding to the empty Coxeter graph and the trivial Coxeter group.

Many of the results and definitions in this paper are valid in all types $A, B, D$ , with definitions for type $A_{n-1}$ naturally fitting with definitions in types $B_n, D_n$ . In this case, we will write ‘type $X_n$ , where $X$ is $A_{-1}$ , $B$ or $D$ ’ (or something in a similar vein).

2.2. Iwahori–Hecke algebras

Let $\mathcal{A} = \mathbb{Q}(v)$ . The Iwahori–Hecke algebra $\mathcal{H}_v(W,S) = \mathcal{H}(W,S)$ associated with a Coxeter system $(W,S)$ is a unital $\mathcal{A}$ -algebra with generators $t_s, s \in S,$ subject to the relations:

1. (i) $\langle t_{s_i},t_{s_j}\rangle _{m_{ij}} = \langle t_{s_j},t_{s_i}\rangle _{m_{ij}}$ , where $\langle x,y \rangle _{m}$ is an alternating product $xyx\ldots$ of $x$ and $y$ with $m$ factors.

2. (ii) $(t_s - v)(t_s + v^{-1}) = 0.$

When $(W,S)$ is of type $B_n$ , we will also be interested in the Iwahori–Hecke algebra $\mathcal{H}_{v,v_0}(W,S)$ with unequal parameters $v, v_0 \in \mathcal{A}$ . This is a unital algebra over the ring $\mathcal{A}$ , with the relation (ii) replaced by the relations:

1. (ii^′) $(t_{s_i} - v)(t_{s_i} + v^{-1}) = 0$ for $i \gt 0$ ,

2. (ii^′) $(t_{s_0} - v_0)(t_{s_0} + v_{0}^{-1}) = 0$ .

When $(W,S)$ is of type $X_n$ with $X \in \{A, B=C, D\}$ , we will write $W(X_n)$ and $\mathcal{H}(X_n)$ for $W$ and $\mathcal{H}(W,S)$ , or $\mathcal{H}_{v,v_0}(B_n)$ if we need to emphasise that we use unequal parameters. In each of these cases, we will write $t_i$ for $t_{s_i}$ .

Let $w = s_{i_1}\ldots s_{i_r}, w \in W, s_{i_k} \in S$ be a reduced expression. Then the element $t_w = t_{s_{i_1}}\ldots t_{s_{i_r}}$ in $\mathcal{H}(W,S)$ is independent of the choice of a reduced expression, and $\mathcal{H}(W,S)$ is a free $\mathcal{A}$ -module with a basis $t_w, w \in W$ . The elements $t_w$ are invertible, and the collection $t_{w}^{-1}, w \in W,$ gives another basis of $\mathcal{H}(W,S)$ over $\mathcal{A}$ .

2.3. Trace functional and a pairing on $\mathcal{H}(W,S)$

Let $x \mapsto \overline {x}$ be the Kazhdan–Lusztig involution on $\mathcal{H}(W,S)$ . It is a $\mathbb{Q}$ -algebra involution, defined by the assignments $\overline {v} = v^{-1}, \overline {t_w} = t_{w^{-1}}^{-1}.$ Let $\omega \,: \, \mathcal{H}(W,S)$ stand for the $\mathcal{A}$ -linear algebra anti-involution defined by $\omega (t_w) = t_{w^{-1}}$ .

For $h \in \mathcal{H}(W,S), x \in W,$ let $h_x$ stand for the $t_x^{-1}$ -coefficient of $h$ in the basis $t_w^{-1},$ where $w$ runs through $W$ . Let $\tau \, : \, \mathcal{H}(W,S) \to \mathcal{A}$ be a functional defined by $\tau (h) = h_e,$ where $e \in W$ is the unit. For $h, h' \in \mathcal{H}(W,S)$ , define a pairing $\langle h, h' \rangle = \tau (\omega (\overline {h})h')$ . We have the following standard proposition, see, for example, [Reference Geck and Pfeiffer8, Chapter 8].

We will use the following straightforward corollary of this proposition repeatedly.

2.4. ‘Full twist’ and Jucys–Murphy elements

Assume that $(W, S)$ is of finite type. Let $w_0$ stand for the longest element in $W$ with respect to the Bruhat length. Let $\mathbb{S}(W,S) = t_{w_0}^{-2} \in \mathcal{H}(W,S)$ . In classical types, we will write $\mathbb{S}(X_n) \in \mathcal{H}(X_n)$ , when $X$ is $A, B$ or $D$ , or simply $\mathbb{S}$ when the type is clear from the context.

The following proposition is also standard:

We will need the following ‘Serre property’ of the operator of multiplication by $\mathbb{S}(W,S)$ .

Proposition 2.4.2. For any $h_1, h_2 \in \mathcal{H}(W,S)$ , we have the identity

\begin{equation*} \langle h_1, \mathbb{S}(W,S)h_2 \rangle = \overline {\langle h_2, h_1 \rangle }. \end{equation*}

Proof. By Proposition 2.3.1 (b), it is enough to prove the statement for $h_1 = t_{w_1}^{-1}, h_2 = t_{w_2}$ for arbitrary $w_1, w_2 \in W$ . We have, by Proposition 2.3.1 (c),

\begin{align*} \langle t_{w_1}^{-1}, \mathbb{S}(W,S)t_{w_2} \rangle &= \langle t_{w_0}t_{w_1}^{-1}, t_{w_0}^{-1}t_{w_2}\rangle = \langle t_{w_0w_1^{-1}}, t_{w_2^{-1}w_0}^{-1} \rangle \\ &= \delta _{w_0w_1^{-1},w_0w_2} = \delta _{w_1,w_2^{-1}} = \overline {\langle t_{w_2}, t_{w_1}^{-1}\rangle }. \end{align*}

This completes the proof.

We have the natural embedding $\iota _n \, : \,\mathcal{H}(X_n) \to \mathcal{H}(X_{n+1})$ , where $X$ is $A, B$ or $D$ , induced by the embedding of the sets of labels of the Coxeter diagram. Using these embeddings, an element of $\mathcal{H}(X_n)$ can be considered as an element of $\mathcal{H}(X_m)$ for $m \gt n$ .

We have the following corollary of Proposition 2.4.1.

2.5. Some central elements in the classical Iwahori–Hecke algebras

In this section, we follow [Reference Ram and Ramagge17]. Let $\mathcal{H}^{\mathrm{aff}}(A_{n-1})$ be the affine Hecke algebra associated with the root datum of $\mathrm{GL}_n$ . It is a unital algebra over $\mathcal{A}$ generated by the elements $t_1, \ldots , t_{n-1}$ and $\Theta _1^{\pm 1}$ , where $t_1, \ldots , t_{n-1}$ satisfy the relations of the Iwahori–Hecke algebra of type $A_{n-1}$ and, additionally, the following relations hold:

1. (iii) $t_1\Theta _1 t_1\Theta _1 = \Theta _1 t_1 \Theta _1 t_1$ ,

2. (iv) $t_i\Theta _1 = \Theta _1 t_i$ for $i \gt 1$ .

Inductively define $\Theta _{i} = t_{i-1}\Theta _{i-1}t_{i-1} \in \mathcal{H}^{\mathrm{aff}}(A_{n-1}), i = 2, \ldots , n$ . It is well known that $\Theta _i^{\pm 1}$ generate a commutative subalgebra of $\mathcal{H}^{\mathrm{aff}}(A_{n-1})$ , called the Bernstein subalgebra. We have the following particular case of a classical theorem, proved in [Reference Lusztig14, Theorem 8.1] for all affine types.

Fix $n \geq 2$ . Note that

1. (i) The Hecke algebra $\mathcal{H}(A_{n-1})$ is the quotient of the affine Hecke algebra $\mathcal{H}^{\mathrm{aff}}(A_{n-1})$ by the relation $\Theta _1 = 1$ .

2. (ii) The Hecke algebra $\mathcal{H}_{v,v_0}(B_n)$ is the quotient of $\mathcal{H}^{\mathrm{aff}}(A_{n-1})$ by the relation

   \begin{equation*}(\Theta _1 - v_0)(\Theta _1 + v_0^{-1}) = 0.\end{equation*}
   Here, $t_0$ is the image of $\Theta _1$ .

3. (iii) The Hecke algebra $\mathcal{H}(D_n)$ is the subalgebra of the quotient (ii) with $v_0 = 1$ , generated by the image $t_{1'}$ of $\Theta _1 t_1 \Theta _1^{-1}$ and images of $t_1, \ldots , t_{n-1}$ .

Note that this exhibits $\mathcal{H}(D_n)$ as a subalgebra of $\mathcal{H}_{v,1}(B_n)$ . Note also that $J^A_i$ is the image of $\Theta _i$ in $\mathcal{H}(A_{n-1})$ for $i = 1, \ldots , n$ . We have

\begin{equation*}J^A_{i} = t_{i-1}\ldots t_1t_1 \ldots t_{i-1}.\end{equation*}

We have the following

3.1. Markov traces via Jucys–Murphy elements

Let $\mathcal{R} = \mathcal{A}(a)$ and let $\overline {a} = a^{-1}$ . Define

\begin{equation*} \zeta _n = \prod _{i = 1}^{n} (1 + a^{-1}J^X_i), \zeta _0 = 1, \end{equation*}

an element in $\mathcal{H}(X_n)$ when $X = A_{-1}, B$ or $D$ . By Corollary 2.5.3, elements $\zeta _n$ are central in the corresponding Iwahori–Hecke algebras. We prove

Proof. We first check the property (U1). We have

\begin{equation*} \langle \zeta _n, \iota (h)\rangle = \langle \zeta _{n-1}(1+a^{-1}J^X_n), \iota (h) \rangle = \langle \zeta _{n-1}, \iota (h)\rangle + a\langle \zeta _{n-1}J_n^X, \iota (h)\rangle . \end{equation*}

Applying Proposition 2.4.2 twice – recall that $J_n^X = \mathbb{S}(X_n)^{-1}\mathbb{S}(X_{n-1})$ – we get the equality $\langle \zeta _{n-1}J_n^X, \iota (h)\rangle = \langle \zeta _{n-1}, \iota (h)\rangle$ and (U1) follows.

We now check the property (M2). We have

(2) \begin{align} \langle \zeta _n, \iota (h)t_{n-1}\rangle &= \langle \zeta _{n-1}(1+a^{-1}J^X_n), \iota (h)t_{n-1} \rangle = \langle \zeta _{n-1}, \iota (h)t_{n-1}\rangle + a\langle \zeta _{n-1}J_n^X, \iota (h)t_{n-1}\rangle \\ \nonumber &= \langle \zeta _{n-1}, \iota (h)t_{n-1}^{-1}\rangle +(v-v^{-1}) \langle \zeta _{n-1}, \iota (h)\rangle + a\langle \zeta _{n-1}J_n^X, \iota (h)t_{n-1}\rangle . \end{align}

From Corollary 2.3.2, we get that $\langle \zeta _{n-1},\iota (h)t_{n-1}^{-1}\rangle = 0.$

For the third summand in the last expression in (2), applying Proposition 2.4.2, we have

\begin{equation*} \langle \zeta _{n-1}J_n^X, \iota (h)t_{n-1}\rangle = \overline {\langle \iota (h)t_{n-1},\zeta _{n-1}\mathbb{S}(X_{n-1})\rangle } = 0,\end{equation*}

with the last equality again following from Corollary 2.3.2. We obtain (M2).

3.2. General Markov traces in types $B, D$

In type $A$ , the classical result of Jones and Ocneanu [Reference Jones10] says that a Markov trace with given constants $\rho , \mu$ is defined uniquely up to an overall scaling. Thus, Theorem3.1.4 gives a general description of a Markov trace in type $A$ .

In types $B, D$ , the general description of Markov traces was obtained by Geck and Lambropoulou [Reference Geck3, Reference Geck and Lambropoulou5]. We recall their results.

For $n \geq 1$ , define the elements

\begin{equation*}T_n = t_{n-1}\ldots t_1 t_0 t_1^{-1}\ldots t_{n-1}^{-1}\in \mathcal{H}(B_n), n \geq 2, T_1 = t_0,\end{equation*}

and

\begin{equation*}U_n = t_{n-1}\ldots t_{1'} t_1^{-1}\ldots t_{n-1}^{-1}\in \mathcal{H}(D_n), n \geq 2, U_1 = 1.\end{equation*}

We will be interested in a particular case of these traces when $y_n$ is of the form $y_n = y^n.$ Let from now on $\mathcal{R} = \mathcal{A}(y)$ and denote by $\mathrm{tr}^{X}_{n,y}$ the Markov trace from Theorem3.2.1 with $y_n = y^n$ for all $n \geq 1$ . Extend the bar involution to $\mathcal{R}$ , formally writing $\overline {y}$ — we assume that such an extension exists under any specialisation of $y$ we will use. To simplify the formulas, we adopt the notation $\alpha = v - v^{-1}, \alpha _0 = v_0 - v_0^{-1}$ . Define the elements

\begin{equation*} \beta _n(y) = \prod _{i = 1}^n(1 + (\overline {y}+\alpha _0)j^B_i + a^{-1}J^B_i) \in \mathcal{H}_{v,v_0}(B_n) \otimes _{\mathcal{A}}\mathcal{R}. \end{equation*}

By Corollary 2.5.3, the element $\beta _n(y)$ is central in $\mathcal{H}_{v,v_0}(B_n)\otimes _{\mathcal{A}}\mathcal{R}$ . We have the following

We will use the following lemma, verified by a direct computation.

The property in Theorem3.2.1 (B) follows from the more general property (compare [Reference Geck and Lambropoulou5, Proposition 4.5]).

Proof. To simplify the formulas, introduce the notation $y' = y-\alpha _0, \overline {y'} = \overline {y}+\alpha _0$ . We have

\begin{equation*} \langle \beta _n(y), \iota (h)T_n\rangle = \langle \beta _{n-1}(y), \iota (h)T_n\rangle + {y'}\langle \beta _{n-1}(y)j^B_n,\iota (h)T_n\rangle +a\langle \beta _{n-1}(y)J_n^B, \iota (h)T_n\rangle . \end{equation*}

Let us examine the summands on the right-hand side of this equation one by one. We will routinely use Corollary 2.3.2. For the first summand, we have, by Lemma 3.2.3 (a), (b),

\begin{equation*} \langle \beta _{n-1}(y), \iota (h)T_n\rangle = \langle \beta _{n-1}(y), \iota (h)T_{n}^{-1}\rangle + \alpha _0\langle \beta _{n-1}(y), \iota (h)\rangle = \alpha _0\langle \beta _{n-1}(y), \iota (h)\rangle . \end{equation*}

For the second summand, first note that $j^B_n$ is in the centraliser of $\mathcal{H}(B_{n-1})$ in $\mathcal{H}(B_n)$ . It follows that

\begin{equation*} {y'}\langle \beta _{n-1}(y)j^B_n,\iota (h)T_n\rangle = {y'}\langle \beta _{n-1}(y),\iota (h)(j_n^B)^{-1} T_n\rangle . \end{equation*}

Thus, we have, by Lemma 3.2.3 (c),

\begin{equation*} {y'}\langle \beta _{n-1}(y),\iota (h)(j^{B}_n)^{-1}T_n\rangle = y'\langle \beta _{n-1}(y), \iota (h)\rangle . \end{equation*}

For the third summand, by Proposition 2.4.2 and Lemma 3.2.3 (b), we have

\begin{equation*} a\langle \beta _{n-1}(y)J_n^B, \iota (h)T_n\rangle = \overline {\langle \iota (h)T_n, \beta _{n-1}(y)\mathbb{S}(B_{n-1})\rangle } = 0. \end{equation*}

Collecting all three summands together, we get

\begin{equation*} \langle \beta _n(y), \iota (h)T_n\rangle = (\alpha _0+y')\langle \beta _{n-1}(y),\iota (h)\rangle = y\langle \beta _{n-1}(y),\iota (h)\rangle , \end{equation*}

which completes the proof.

Proof of Theorem 3.2.2.It remains to check the properties (U1) and (M2). Note that the defining expression (1) for $j^B_n$ is reduced as a word in generators $t_i$ and ends in $t_{n-1}$ , so $j^B_n = t_{c}, j^B_nt_{n-1}^{-1} = t_{c'}$ for some $c, c' \in W(B_n)\backslash W(B_{n-1})$ . It follows from Corollary 2.3.2 that, for any $h\in \mathcal{H}(B_{n-1})$ we have

\begin{equation*} \langle (\overline {y}+\alpha _0)\beta _{n-1}j_n^{B}, \iota (h) \rangle = 0 = \langle (\overline {y}+\alpha _0)\beta _{n-1}j_n^{B}, \iota (h)t_{n-1}\rangle , \end{equation*}

which reduces the proof to the proof of Theorem3.1.4.

From Theorems3.2.1 and 3.2.2, we get the following description of a Markov trace $\mathrm{tr}^D_{n,y}$ .

Corollary 3.2.5. Let

\begin{equation*}P_n(x_1, \ldots , x_n; \,y) = \prod _{i=1}^n(1 + \overline {y}x_i + a^{-1}x_i^2),\end{equation*}

a symmetric polynomial in the variables $(x_i)$ over $\mathcal{R}$ . Write $P_n = P_n^+ + P_n^-$ , a sum of even and odd monomial degree parts. Let $\delta _n(y) = P_n^+(j_1^D, \ldots , j_n^D; \, y)$ . Then $\mathrm{tr}_{n,y}^D = \mathrm{tr}_{\delta _n(y)}$ .

Proof. Recall that the trace $\mathrm{tr}_{n,(y_{\bullet })}^D$ is obtained by restriction of $\mathrm{tr}_{n,(y_{\bullet })}^B$ along the embedding $\iota _{D\to B}\, : \, \mathcal{H}(D_n) \to \mathcal{H}_{v,1}(B_n)$ , so that we have

\begin{equation*} \mathrm{tr}_{n,y}^D(h) = \langle \beta _n(y),\iota _{D\to B}(h)\rangle = \langle P_n(j_1^B,\ldots ,j_n^B; \, y),\iota _{D\to B}(h)\rangle . \end{equation*}

It remains to observe that for all $i$ we have $j_i^D = t_0j_i^B$ , and so every even degree monomial in $j_i^B$ is equal to the same monomial in $j_i^D$ , since $t_0^2 = 1$ in $\mathcal{H}_{v,1}(B_n)$ and $t_0$ commutes with $j_i^B$ for all $i$ ; and that every odd degree monomial in $j_i^B$ is orthogonal to $\iota _{D\to B}(h)$ with respect to $\langle \,,\rangle$ . This completes the proof of the Corollary.

We have the following analogue of Proposition 3.2.4 in type $D$ .

Proposition 3.2.6. For $n \geq 1$ and any $h \in \mathcal{H}(D_{2n-2})$ , we have

\begin{equation*}\mathrm{tr}_{2n,y}^D(\iota \circ \iota (h)U_{2n-1}U_{2n}) = y^2\mathrm{tr}_{2n-2,y}^D(h).\end{equation*}

Proof. Note that we have $\iota _{D \to B}(U_k) = t_0T_k$ , and so

\begin{equation*}\mathrm{tr}_{2n,y}^D(\iota \circ \iota (h)U_{2n-1}U_{2n}) = \mathrm{tr}_{2n,y}^B(\iota \circ \iota (h)t_0T_{2n-1}t_0T_{2n}).\end{equation*}

Using the property (M1) and Proposition 3.2.4 twice, we get the result.

3.3. Geometric Markov traces

Markov traces $\mathrm{tr}_n^B\, : \, \mathcal{H}_{v,v}(B_n) \to \mathcal{A}(a), \mathrm{tr}_n^D\, : \, \mathcal{H}(D_n) \to \mathcal{A}(a)$ studied in [Reference Gomi7, Reference Webster and Williamson20] are specialisations of traces $\mathrm{tr}_{n,y}^X$ satisfying the extension of the properties (U1) and (M2) for the embeddings $B_0 \to B_1$ and $D_1 \to D_2, D_0 \to D_1$ . Since these traces are of geometric origin, we call them ‘geometric Markov traces’. In particular, these traces satisfy

\begin{equation*} \mathrm{tr}_1^B(t_0) = v-v^{-1}, \mathrm{tr}_1^D(t_1) = v-v^{-1}, \mathrm{tr}^D_2(t_1t_{1'}) = (v-v^{-1})^2. \end{equation*}

The condition on $\mathrm{tr}_1^B$ forces $y = v - v^{-1}$ and $\overline {y}+\alpha _0 = 0$ . Thus, we get that $\mathrm{tr}_n^B = \mathrm{tr}_{\zeta _n}$ , the trace from Theorem3.1.4.

The conditions on $\mathrm{tr}_1^D, \mathrm{tr}_2^D$ imply that

\begin{equation*} y^2 = \mathrm{tr}_2^D(U_1U_2) = \mathrm{tr}_2^D(t_{1'}t_{1}^{-1}) = \mathrm{tr}_2^D(t_1t_{1'}) - \alpha \mathrm{tr}_2^D(t_{1'}) = \alpha ^2 - \alpha ^2(1+a) = -\alpha ^2a. \end{equation*}

It is easy to see that geometric Markov traces $\mathrm{tr}_n^B, \mathrm{tr}_n^D$ take values in $\mathcal{A}[a]$ . Let $\mathrm{tr}_n^{X,k}\, : \, \mathcal{H}(X_n) \to \mathcal{A}$ be the coefficient of $\mathrm{tr}_n^X$ at $a^k$ . We summarise the above discussion in the following

Theorem 3.3.1. The functional $\mathrm{tr}_n^{B,k}$ is represented by the $k$ th elementary symmetric polynomial in Jucys–Murphy elements:

\begin{equation*} \mathrm{tr}_n^{B,k}(h) = \langle e_k(J_1^B, \ldots , J_n^B), h\rangle . \end{equation*}

Let $e_k'(x_1, \ldots , x_n; \, \alpha )$ be the symmetric polynomial in $(x_i)$ appearing as a degree $2k$ homogeneous part in

\begin{equation*} \prod _{i=1}^n(1 + \alpha x_i - x_i^2) = \prod _{i=1}^n(v-x_i)(v^{-1}+x_i), \end{equation*}

where $\alpha = v - v^{-1}$ . Then

\begin{equation*} \mathrm{tr}_n^{D,k}(h) = ( \! -1)^k\langle e'_k(j_1^D, \ldots , j_n^D; \, \alpha ),h \rangle . \end{equation*}

The symmetric polynomial $e_k'$ can be written explicitly as

\begin{equation*} e_k' = \sum _{\substack {i+j = 2k,\\ i,j\geq 0}}(\!-1)^{i}v^{j-i}e_ie_j. \end{equation*}

Note that the highest $a$ -coefficient is always represented by the ‘full twist’ element $\mathbb{S}^{-1} = e_n(J_1^X, \ldots , J_n^X), \mathrm{tr}_n^{X,n} = \mathrm{tr}_{\mathbb{S}^{-1}}.$ This recovers the symmetry of the HOMFLY-PT polynomial observed by Kálmán [Reference Kálmán11] that was categorified in [Reference Gorsky, Hogancamp, Mellit and Nakagane4] in type $A$ and in [Reference Bezrukavnikov and Tolmachov2] in general type.