Kazhdan-Lusztig polynomials and subexpressions

Let $(W,S)$ be a Coxeter group with length function $\ell$ and Bruhat order $\leq$. An expression $\underline{x}=(s_{1},s_{2},\dots,s_{m})$ is a word in the alphabet $S$ (i.e. $s_{i}\in S$ for all $i$). Its length is $\ell(\underline{x})=m$.

If $\underline{x}=(s_{1},s_{2},\dots,s_{m})$ is an expression, we let $x:=s_{1}s_{2}\dots s_{m}$ denote the product in $W$. Given an expression $\underline{x}=(s_{1},s_{2},\dots,s_{m})$, a subexpression of $\underline{x}$ is a word $\underline{e}=e_{1}e_{2}\dots e_{m}$ of length $m$ in the alphabet $\{0,1\}$. We will write $\underline{e}\subset\underline{x}$ to indicate that $\underline{e}$ is a subexpression of $\underline{x}$. We set

and say that $\underline{e}\subset\underline{x}$ expresses $\underline{x}^{\underline{e}}$.

For $1\leq i\leq m$, we define $w_{i}:=s_{1}^{e_{1}}s_{2}^{e_{2}}\dots s_{i}^{e_{i}}$. We also define $d_{i}\in\{U,D\}$ (where $U$ stands for Up and $D$ for Down) in the following way:

We write the decorated sequence $(d_{1}e_{1},\ldots,d_{m}e_{m})$. Deodhar’s defect $\operatorname{df}$ is defined by

For the basic definitions of Hecke algebras and Kazhdan-Lusztig polynomials we follow [Soe97]. Let $(W,S)$ be a Coxeter system. Recall that the Hecke algebra $\mathcal{H}$ of $(W,S)$ is the algebra with free $\mathbb{Z}[v,v^{-1}]$-basis given by symbols $\{h_{x}\}_{x\in W}$ and multiplication given by

We can define a $\mathbb{Z}$-module morphism $\overline{(-)}:\mathcal{H}\rightarrow\mathcal{H}$ by the formula $\overline{v}=v^{-1}$ and $\overline{h_{x}}=(h_{x^{-1}})^{-1}.$ It is a ring morphism, and we call it the duality in the Hecke algebra. The Kazhdan-Lusztig basis of $\mathcal{H}$ is denoted by $\{b_{x}\}_{x\in W}$. It is a $\mathbb{Z}[v,v^{-1}]-$basis of $\mathcal{H}$ and it is characterised by the two conditions

for all $x\in W$. If we write $b_{x}=h_{x}+\sum_{y\in W}h_{y,x}h_{y}$ then the Kazhdan-Lusztig polynomials (as defined in [KL79]) $p_{y,x}$ are defined by the formula $p_{y,x}=v^{l(x)-l(y)}h_{y,x},$ and $C^{\prime}_{x}=b_{x}$ (their $q^{-1/2}$ is our $v$).

Let us define the $\mathbb{Z}[v,v^{-1}]$-bilinear form

given by $(h_{x},h_{y}):=\delta_{x,y}$. A useful property of this pairing is that $(b_{x},b_{y})\in v\mathbb{Z}[v]$ if $x\neq y$ and $(b_{x},b_{x})\in 1+v\mathbb{Z}[v]$.

We fix a realisation ${\mathfrak{h}}$ of our Coxeter system $(W,S)$ over the real numbers ${\mathbb{R}}$. That is, ${\mathfrak{h}}$ is a real vector space and we have fixed roots $\{\alpha_{s}\}_{s\in S}\subset{\mathfrak{h}}^{*}$ and coroots $\{\alpha_{s}^{\vee}\}_{s\in S}\subset{\mathfrak{h}}$ such that the familiar formulas from Lie theory define a representation of $W$ of ${\mathfrak{h}}$ and ${\mathfrak{h}}^{*}$.

Throughout, we assume that this is a realisation for which Soergel’s conjecture holds. For example we could take ${\mathfrak{h}}$ to be the realisation from [Soe07, EW14]. We could also take ${\mathfrak{h}}$ to be the geometric representation [Lib08a] so that ${\mathfrak{h}}=\bigoplus{\mathbb{R}}\alpha_{s}^{\vee}$ and for $t\in S,$ the element $\alpha_{t}\in{\mathfrak{h}}^{*}$ is defined by $\langle\alpha_{t},\alpha_{s}^{\vee}\rangle=-\cos(\pi/m_{st})$, where $m_{st}$ denotes the order (possibly $\infty$) of $st\in W$.

Having fixed ${\mathfrak{h}}$ we define $R=S({\mathfrak{h}}^{*})=\mathcal{O}({\mathfrak{h}})$ to be the symmetric algebra on ${\mathfrak{h}}^{*}$ (alias the polynomials functions on ${\mathfrak{h}}$), graded so that ${\mathfrak{h}}^{*}$ has degree 2. We denote by $\textrm{Bim}_{R}$ the category of $\mathbb{Z}-$graded $R$-bimodules which are finitely generated both as left and right $R$-modules. Given an object $M=\bigoplus M^{i}\in\textrm{Bim}_{R}$ we denote by $M(k)$ the shifted bimodule, with $M(k)^{i}:=M^{k+i}$. Given objects $M,N\in\textrm{Bim}_{R}$ we denote their tensor product by juxtaposition: $MN:=M\otimes_{R}N$. This operation makes $\textrm{Bim}_{R}$ into a monoidal category. The Krull-Schmidt theorem holds in $\textrm{Bim}_{R}$.

For any $s\in S$ we denote by $R^{s}\subset R$ the $s$-invariants in $R$. We consider the bimodule

Given an expression $\underline{w}=(s_{1},\dots,s_{m})$ we consider the Bott-Samelson bimodule

The category ${\mathcal{B}}$ of Soergel bimodules is defined to be the full, strict (i.e. closed under isomorphism), additive (i.e. $M,N\in{\mathcal{B}}\Rightarrow M\oplus N\in{\mathcal{B}}$), monoidal (i.e. $M,N\in{\mathcal{B}}\Rightarrow MN\in{\mathcal{B}}$) category of $\textrm{Bim}_{R}$ which contains $B_{s}$ for all $s\in S$ and is closed under shifts $(m)$ and direct summands.

Soergel proved the following facts (usually known as Soergel’s categorification theorem). For all $w\in W$ there exists a unique (up to isomorphism) bimodule $B_{w}$ which occurs as a direct summand of $B_{\underline{w}}$ for any reduced expression $\underline{w}$ of $w$, and is not a summand of (some shift of) $B_{\underline{y}}$ for any shorter sequence $\underline{y}$. The set $\{B_{w}\,|\,w\in W\}$ constitutes a complete set of non-isomorphic indecomposable Soergel bimodules, up to isomorphism and grading shift. There is a unique isomorphism of $\mathbb{Z}[v,v^{-1}]$-algebras between the split Grothendieck group of ${\mathcal{B}}$ and the Hecke algebra

satisfying $\mathrm{ch}([B_{s}])=b_{s}$ and $\mathrm{ch}([R(1)])=v$.

Soergel gave a formula to calculate the graded dimensions of the Hom spaces in ${\mathcal{B}}$ in the Hecke algebra. We need some notation to explain it. Given a finite dimensional graded $\mathbb{R}$-vector space $V=\oplus V^{i}$, we define

Given a finitely-generated and free graded right $R$-module $M$, we define

The following is Soergel’s hom formula. Let $M,N\in{\mathcal{B}}$, then $\operatorname{Hom}(M,N)$ is finitely-generated and free as a right $R$-module, and

Soergel’s conjecture (now a theorem by Elias and the second author [EW14]) is the following statement:

We remark that when Soergel’s conjecture is satisfied (the case considered in this paper), by Soergel’s hom formula and by the useful property at the end of Section 2.2, we obtain a complete description of the degree zero morphisms between indecomposable objects:

An important result in the theory of Soergel bimodules is a theorem of the first author giving a “double leaves” basis of morphisms between Soergel bimodules. Let $\underline{w}=(s_{1},\dots,s_{m})$ denote an expression. For any subexpression $\underline{e}$ of $\underline{w}$ the first author associates a morphism

Here $\underline{x}$ is a fixed but arbitrary reduced expression of $x=\underline{w}^{\underline{e}}$. The definition of $\operatorname{LL}_{\underline{w},\underline{e}}$ is inductive, and will not be given here, as we will not need it. However it is important to note that the definition of $\operatorname{LL}_{\underline{w},\underline{e}}$ depends on choices (fixed reduced expressions for elements and fixed sequences of braid relations between reduced expressions) which seem difficult to make canonical.

However, once one has fixed such choices one can produce a basis of homomorphisms between any two Bott-Samelson bimodules. Indeed, a theorem of the first author [Lib15, Thm. 3.2] (see also [EW16, Thm 6.11]) asserts that the set

is a free $R$-basis for $\mathrm{Hom}(B_{\underline{z}},B_{\underline{w}})$.

Let $M,N\in{\mathcal{B}}$. For $x\in W$ we denote by

the vector space generated by all morphisms $f:M\rightarrow N$ that factor through $B_{y}(n)$ for some $y<x$ and $n\in\mathbb{Z}$. Let

We denote by ${\mathcal{B}}_{\not<x}$ the category whose objects coincide with those of ${\mathcal{B}}$ and for any $M,N\in{\mathcal{B}}_{\not<x}$ we have $\operatorname{\mathrm{Hom}}_{{\mathcal{B}}_{\not<x}}(M,N):=\operatorname{\mathrm{Hom}}_{\not<x}(M,N)$.

Consider the sets

The set $D_{x,y}$ is a canonical direct summand of $D_{x,\underline{y}}$. This is because, when Soergel’s conjecture is satisfied, there is one element in $\mathrm{End}(B_{\underline{y}})$ projecting to $B_{y}$ called the favorite projector (see [Lib15, §4.1]). Let us give the construction of this projector. Let us assume (by induction) that projection and inclusion maps have been constructed

for some reduced expression $\underline{y}$ of $y$. Suppose $y<ys,$ then

By Soergel’s conjecture this implies

By (2.1), there is only one projector in this space projecting to $B_{ys}$, which we write as

We now define the inclusion and projection maps of our favourite projector to be the compositions

What do we really mean when we write $B_{x}$? In the general setting of Soergel bimodules, we mean a representative of an equivalence class of isomorphic bimodules, where each isomorphism is not canonical. In our setting (where Soergel’s conjecture is available), we mean a representative of an equivalence class of isomorphic bimodules, where each isomorphism is canonical up to an invertible scalar (in our case ${\mathbb{R}}^{*}$). We now explain a somewhat adhoc way to fix this scalar, so that $B_{x}$ is defined up to unique isomorphism.

Consider an expression $\underline{x}$, and the corresponding Bott-Samelson bimodule $B_{\underline{x}}$. It contains a canonical element

(Note that $B_{\underline{x}}$ is zero below degree $-\ell(\underline{x})$ and is spanned by $c_{\textrm{bot}}$ in degree $-\ell(\underline{x})$; bot stands for “bottom”.) We denote by $c_{\textrm{bot}}^{x}\in B_{x}$ the image of $c_{\textrm{bot}}^{\underline{x}}$ under the favourite projector, where $\underline{x}$ is a reduced expression for $x$.

From now on we will always understand $B_{x}$ to mean $B_{x}$ together with the element $c_{\textrm{bot}}\in B_{x}$. Given two representatives $(B_{x},c_{\textrm{bot}}^{x})$ and $(\tilde{B}_{x},\tilde{c}_{\textrm{bot}}^{x})$, there is a unique isomorphism $B_{x}\rightarrow\tilde{B}_{x}$ which sends $c_{\textrm{bot}}^{x}$ to $\tilde{c}_{\textrm{bot}}^{x}$.

In this section we introduce the canonical maps which will be our building blocks for the definition of canonical light leaves, in the next section.

Let $x\in W$ and $s\in S$ be as in the lemma above (i.e. $x<xs$). Both $B_{x}B_{s}$ and $B_{xs}$ are one-dimensional in degree $-\ell(x)-1$, where they are spanned by $c_{\textrm{bot}}^{x}c_{\textrm{bot}}^{s}$ and $c_{\textrm{bot}}^{xs}$ respectively. (We write $c_{\textrm{bot}}^{x}c_{\textrm{bot}}^{s}$ instead of $c_{\textrm{bot}}^{x}\otimes c_{\textrm{bot}}^{s}$.) Hence there exists a unique map

which maps $c_{\textrm{bot}}^{x}c_{\textrm{bot}}^{s}$ to $c_{\textrm{bot}}^{xs}$. Similar considerations show that there exists a unique map

resp.

mapping $c_{\textrm{bot}}^{xs}c_{\textrm{bot}}^{s}$ to $c_{\textrm{bot}}^{xs}$ (resp. $c_{\textrm{bot}}^{xs}$ to $c_{\textrm{bot}}^{x}$).

We will use the maps $\alpha_{x,s},\beta_{x,s}$ and $\gamma_{x,s}$ constructed above. We will also use the multiplication map

Consider the following data:

1. (1)

   an expression (not necessarily reduced) $\underline{y}=(s_{1},\ldots,s_{n})$;

2. (2)

   elements $x\in W$, $s\in S$; and

3. (3)

   $f:B_{\underline{y}}\rightarrow B_{x}$.

To this data, we will associate two new maps:

These maps are constructed as follows: If $x<xs$, define

If $xs<x$, define

Given an expression $\underline{w}$ and a subexpression $\underline{e}$ define the canonical light leaf

where $\mathrm{id}$ means $\mathrm{id}\in\mathrm{End}(R)$ and for example $\mathrm{id}(0,1,0)$ means $(((\mathrm{id}0)1)0)$.

The proof of the following theorem is essentially the same as in [Lib15, Thm. 3.2] and [EW16, Thm 6.11].

Now we can explain why this theorem proves Theorem 1.1. By Theorem 3.6, the graded set $\{\operatorname{CLL}_{\underline{y},\underline{e}}\,|\,\mathrm{with}\,\underline{e}\,\mathrm{expressing}\,x\}$ is naturally an $R$-basis of $\widehat{D}_{x,\underline{y}}$, thus it gives an $\mathbb{R}$-basis of $D_{x,\underline{y}}.$ So, in summary, the canonical map CLL in Theorem 1.1 is the $\mathbb{R}-$linear map defined on the generators $\underline{e}\in\widetilde{X}_{x,\underline{y}}$ by $\underline{e}\mapsto\mathrm{CLL}_{\underline{y},\underline{e}}.$