Smoothness of Schubert varieties indexed by involutions in finite simply laced types

TL;DR

This paper proves that in finite simply laced types, Schubert varieties indexed by involutions are singular unless they are the longest element of a parabolic subgroup.

Contribution

It establishes a complete characterization of singular Schubert varieties indexed by involutions in finite simply laced types.

Findings

Schubert varieties indexed by involutions are singular unless they are the longest element of a parabolic subgroup

Provides a classification of singularities in Schubert varieties based on involution indexing

Advances understanding of geometric properties of Schubert varieties in algebraic geometry

Abstract

We prove that in finite, simply laced types, every Schubert variety indexed by an involution which is not the longest element of some parabolic subgroup is singular.

Full text

1. Introduction

Let $w$ be an involution in the symmetric group $S_{n}$. In [12] Hohlweg proved that the Schubert variety $X_{w}$ is smooth if and only if $w$ is the longest element of some parabolic subgroup of $S_{n}$. He arrived at this result by exploiting Lakshmibai and Sandhya’s [14] classical pattern avoidance criterion for smoothness of type $A$ Schubert varieties.

The main result of this paper extends Hohlweg’s result to arbitrary finite, simply laced types. Namely, if $W$ is a simply laced Weyl group, and $w\in W$ is an involution, $X_{w}$ is smooth if and only if $w$ is the longest element of a parabolic subgroup of $W$.

It seems likely that it would be possible to arrive at this result in a case by case fashion using the general root system pattern avoidance criteria for smoothness pioneered by Billey and Postnikov [2]. Instead of investigating this approach, we provide a uniform proof based on Carrell-Peterson type criteria in terms of the associated Bruhat graphs.

In Section 2, we recall properties of Coxeter systems and Bruhat graphs which can be used to study smoothness of Schubert varieties in a combinatorial way. In Section 3, we prove Theorem 3.1 which is the main result.

2. Preliminaries

In this section some properties of Bruhat graphs of Coxeter groups are recalled. For more on these concepts, see e.g. [3] and [9].

A Coxeter group is a group $W$ generated by a set $S$ of simple reflections $s$ under relations of the form $s^{2}=e$ and $(ss^{\prime})^{m(s,s^{\prime})}=e$ for all $s,s^{\prime}\in S$ where $e$ is the identity element and $m(s^{\prime},s)=m(s,s^{\prime})$ $\geq 2$ is the order of $ss^{\prime}$ for $s\neq s^{\prime}$. The pair $(W,S)$ is called a Coxeter system, and each element $w\in W$ is a product of generators $s_{i}\in S$, i.e., $w=s_{1}s_{2}\cdots s_{j}$. If $j$ is minimal among all such expressions for $w$, then $j$ is called the length of $w$, denoted $\ell(w)=j$. The Coxeter system $(W,S)$ is simply laced if $m(s,s^{\prime})\leq 3$ for all $s,s^{\prime}\in S$; otherwise it is multiply laced.

If $W$ is finite, there exists a longest element $w_{0}\in W$. It is an involution and satisfies $\ell(v)<\ell(w_{0})$ for all other elements $v\in W$. In fact $w_{0}$ is the unique element in $w$ such that $\ell(sw_{0})<\ell(w_{0})$ for all $s\in S$.

From now on let us fix a Coxeter system $(W,S)$. Let $T=\{wsw^{-1}:w\in W,s\in S\}$ be the set of reflections in $W$. For $v,w\in W$ define:

• (i)

$v\rightarrow w$ if $w=vt$ for some $t\in T$ with $\ell(v)<\ell(w)$.

• (ii)

$v\leq w$ if $v=v_{0}\rightarrow v_{1}\rightarrow\cdots\rightarrow v_{m}=w$ for some $v_{i}\in W$.

The Bruhat graph $\mathrm{Bg}_{S}(W)$ of $(W,S)$ is the directed graph whose vertex set is $W$ and whose edge set is $E_{{}_{\mathrm{Bg}_{S}(W)}}=\{(u,w):u\rightarrow w\}$. The Bruhat order is the partial order relation on $W$ given by (ii).

Example 2.1**.**

Denote by $W(A_{2})$ the Coxeter group of type $A_{2}$ with set of simple reflections $S(A_{2})=\{s_{1},s_{2}\}$ satisfying $m(s_{1},s_{2})=3$. Then $\mathrm{Bg}_{S(A_{2})}(W(A_{2}))$ is as depicted in Figure 1.

The map $v\mapsto v^{-1}$ is an automorphism of the Bruhat order:

Lemma 2.2**.**

For all $v,w\in W$, $v<w$ if and only if $v^{-1}<w^{-1}$.

Define the left descent set of $w$ as $D_{L}(w)=\{s\in S:\ell(sw)<\ell(w)\}$. The following fundamental result about the Bruhat order is sometimes called the Lifting property.

Lemma 2.3** (Verma [16]).**

Suppose $v<w$ and $s\in D_{L}(w)\setminus D_{L}(v)$. Then, $v\leq sw$ and $sv\leq w$.

2.1. Reflection subgroups

Maintain the Coxeter system $(W,S)$ and its set of reflections $T$ as defined above. Then $W^{\prime}$ is a reflection subgroup of $W$ if $W^{\prime}=\langle W^{\prime}\cap T\rangle$. A reflection subgroup $W^{\prime}$ is called dihedral if $W^{\prime}=\langle t,t^{\prime}\rangle$ for some $t,t^{\prime}\in T$, with $t\neq t^{\prime}$.

Lemma 2.4** (Dyer [9]).**

Suppose $t_{1},t_{2},t_{3},t_{4}\in T$ and $t_{1}t_{2}=t_{3}t_{4}\neq e$. Then $W^{\prime}=\langle t_{1},t_{2},t_{3},t_{4}\rangle$ is a dihedral reflection subgroup of $W$.

It turns out that reflection subgroups of $W$ are themselves Coxeter groups. For $w\in W$, define $N(w):=\{t\in T:\ell(tw)<\ell(w)\}$. This is the set of inversions of $w$.

Theorem 2.5** (Deodhar [7], Dyer [8]).**

Let $W^{\prime}$ be a reflection subgroup of $W$ and define $X=\{t\in T:N(t)\cap W^{\prime}=\{t\}\}$. Then,

• (1)

$W^{\prime}\cap T=\{ut^{\prime}u^{-1}:u\in W^{\prime},t^{\prime}\in X\}$.

• (2)

$(W^{\prime},X)$* is a Coxeter system.*

Coxeter described all types of affine groups generated by reflections and their reflection subgroups [6]. The following lemma is a very special case. It can be seen directly e.g. by considering root lengths.

Lemma 2.6**.**

Every reflection subgroup of a finite simply laced group is itself simply laced.

For any subset $Y\subseteq W$ define the Bruhat graph of $Y$, denoted $\mathrm{Bg}_{S}(Y)$, as the directed subgraph of $\mathrm{Bg}_{S}(W)$ induced by $Y$.

Theorem 2.7** (Dyer [9]).**

Let $W^{\prime}$ be a reflection subgroup of $W$ and let $X$ be as in Theorem 2.5. Then $\mathrm{Bg}_{S}(W^{\prime})=\mathrm{Bg}_{X}(W^{\prime})$.

2.2. Schubert varieties

Let $G$ be an algebraic group over $\mathbb{C}$ and $B$ a Borel subgroup containing a maximal torus $T$. Then $G/B$ is called the flag variety and it is the disjoint union of Schubert cells $BwB/B$ where $w\in W$ and $W=N(T)/T$ is the Weyl group (which is a finite Coxeter group). The closure $X_{w}:=\overline{BwB/B}$ is called a Schubert variety. Note that $G/B=X_{w_{0}}$ for $w_{0}$ the longest element of $W$. More on Schubert varieties can be found e.g. in [1].

Next, we review ways to detect singularities of Schubert varieties by inspecting Bruhat graphs. For a lower interval $[e,w]=\{z\in W:e\leq z\leq w\}$ write $\mathrm{Bg}_{S}(w)$ for the Bruhat graph of $[e,w]$. Let $z$ be a vertex in $\mathrm{Bg}_{S}(w)$. The degree of $z$, denoted $\deg_{w}(z)$, is the number of edges incident to $z$ in $\mathrm{Bg}_{S}(w)$ (where directions of edges are ignored).

The following result holds in any Coxeter group. In that generality it is due to Dyer [10]. In our context, where $W$ is a (finite) Weyl group, other proofs given by Carrell and Peterson [4] and Polo [15] also apply.

Theorem 2.8**.**

Let $w\in W$. Then the degree of any vertex in $\mathrm{Bg}_{S}(w)$ is at least $\ell(w)$.

In any Bruhat graph $\mathrm{Bg}_{S}(w)$, it is known that $\ell(w)=|N(w)|=\deg_{w}(w)$. In particular, if $\mathrm{Bg}_{S}(w)$ is regular (i.e., every vertex of $\mathrm{Bg}_{S}(w)$ has the same number of edges), then $\deg_{w}(e)=\ell(w)$.

Theorem 2.9** (Carrell-Peterson [4]).**

The Schubert variety $X_{w}$ is rationally smooth if and only if $\mathrm{Bg}_{S}(w)$ is regular.

Smoothness and rational smoothness are equivalent for simply laced Weyl groups:

Theorem 2.10** (Carrell-Kuttler [5]).**

Suppose $W$ is simply laced. Then for any $w\in W$, $X_{w}$ is smooth if and only if it is rationally smooth.

Corollary 2.11**.**

If $W$ is simply laced then $X_{w}$ is smooth if and only if $\mathrm{Bg}_{S}(w)$ is regular.

In general smoothness is stronger than rational smoothness when $W$ is not simply laced. For example $X_{s_{1}s_{2}s_{1}}$ is rationally smooth but not smooth if $W$ is of type $C_{2}$ generated by the simple reflections $s_{1}$ and $s_{2}$ with $s_{1}$ corresponding to the short root.

The next definition provides another characterization which can be used to prove that a given Schubert variety is not rationally smooth (see Theorem 2.14 below).

Definition 2.12**.**

[13]* Let $x,u,v\leq w$. The Bruhat interval $[e,w]$ contains the broken rhombus $(x,u,v)$ if the conditions below are satisfied:*

• (1)

$x\leftarrow u\rightarrow v$;

• (2)

There is some $y\in W$ with $x\rightarrow y\leftarrow v$;

• (3)

If $x\rightarrow y\leftarrow v$, then $y\nleq w$.

Example 2.13**.**

Consider the group $W(D_{4})$ of type $D_{4}$ with set of simple reflections $S(D_{4})=\{s_{1},s_{2},s_{3},s_{4}\}$ where $m(s_{i},s_{2})=3$ for $i=1,3,4$ and $m(s_{i},s_{j})=2$ for $i,j\neq 2$. In Figure 2 is depicted $\mathrm{Bg}_{S(D_{4})}(w)$ for $w=s_{2}s_{1}s_{3}s_{4}s_{2}$. We use $1,2,3,4$ for $s_{1},s_{2},s_{3},s_{4}$ respectively for brevity. Thus, for example, $w$ is represented by $21342$. The interval $[e,w]$ contains the broken rhombus $(s_{2}s_{3},s_{2},s_{1}s_{2})$ since there is no $y\leq w$ such that $s_{2}s_{3}\rightarrow y\leftarrow s_{1}s_{2}$ although $s_{2}s_{3}\rightarrow s_{1}s_{2}s_{3}\leftarrow s_{1}s_{2}$. Moreover note that $\mathrm{Bg}_{S(D_{4})}(w)$ is not regular. Hence, $X_{w}$ is not smooth for $w=s_{2}s_{1}s_{3}s_{4}s_{2}\in W(D_{4})$.**

The following is [13, Theorem 5.3]. It can also be obtained from the main result of Dyer [11].

Theorem 2.14**.**

The Schubert variety $X_{w}$ is rationally smooth if and only if $[e,w]$ contains no broken rhombus.

Corollary 2.15**.**

Suppose $W$ is simply laced and let $w\in W$. Then, the Schubert variety $X_{w}$ is smooth if and only if $[e,w]$ contains no broken rhombus.

3. Schubert varieties indexed by involutions

In this section, which contains the main result, we will consider Schubert varieties indexed by involutions of finite simply laced groups.

Again let $(W,S)$ be an arbitrary Coxeter system. A parabolic subgroup of $W$ is a subgroup of the form $W_{J}=\langle J\rangle$ for $J\subseteq S$. If $W_{J}$ is finite its longest element will be denoted by $w_{0}(J)$.

For $v\in W$ define $S(v):=\{s\in S:s\leq v\}$. Then $W_{S(v)}$ is the minimal parabolic subgroup of $W$ which contains $v$.

Theorem 3.1**.**

Suppose $(W,S)$ is finite and simply laced and let $v\in W$ be an involution. Then the Schubert variety $X_{v}$ is smooth if and only if $v=w_{0}(J)$ for some $J\subseteq S$.

Proof.

The “if” assertion is obvious: $X_{w_{0}(J)}$ is a (smooth) flag variety. For the “only if” direction, let $v$ be an involution which is not the longest element of any parabolic subgroup $W_{J}$ of $W$. Since $v\neq w_{0}(S(v))$, there exists $s\in S(v)$ such that $v<sv$. If $\deg_{v}(e)\neq\ell(v)$, $\mathrm{Bg}_{S}(v)$ is not regular and, by Corollary 2.11, $X_{v}$ is not smooth. Thus, we may assume $\deg_{v}(e)=\ell(v)$. Since $\deg_{sv}(e)$ is the number of $t\in T$ such that $t\leq sv$ and that degree is at least $\ell(sv)$, there exists a reflection $t\leq sv$ such that $t\nleq v$. Since $t\leq sv$, by Lemma 2.2, $t^{-1}\leq v^{-1}s$ which implies that $t\leq vs$. Also we must have $st<t$. To see this we use Lemma 2.3 in the following way: Since $s\in D_{L}(sv)$, if we would have $t<st$ then by using the lifting property this would imply that $t\leqslant ssv=v$ which is a contradiction. Now since $st<t$, by Lemma 2.2 we see that $ts<t$. Because $ts<t$ and $t\leq vs$, we have $ts<vs$. Since $s\not\in D_{L}(st)$, we have $s\in D_{L}(sv)\setminus D_{L}(st)$ and then by Lemma 2.3 we get $st\leq v$. Consider the dihedral subgroup $D=\langle s,sts\rangle$ of $W$. Since $W$ is finite and simply laced, then by Lemma 2.6, $D$ is also simply laced. Then $(D,X)$ is either of type $A_{1}\times A_{1}$ or of type $A_{2}$, where $X$ is as in Theorem 2.5. By Theorem 2.7 $\mathrm{Bg}_{S}(D)$ equals the Bruhat graph of $(D,X)$. Since, moreover, $st\rightarrow t$, $D$ is not of type $A_{1}\times A_{1}$ and hence $D$ must be of type $A_{2}$. Therefore, $D=\left\{e,s,sts,ts,st,t\right\}$, $\mathrm{Bg}_{S}(D)\cong\mathrm{Bg}_{S(A_{2})}(W(A_{2}))$, and $\mathrm{Bg}_{S}(D)$ is as shown in Figure 3.

Now $ts,st\leq v$ but $t\not\leq v$. From Figure 3, $(st,sts,ts)$ is a broken rhombus of $[e,v]$ because there is no $x\leq v$ such that there are directed edges from $st$ and $ts$ to $x$. To see this, suppose that $st\rightarrow x\leftarrow ts$. So there exist $t^{\prime},t^{\prime\prime}\in T$ with $t^{\prime}\neq t^{\prime\prime}$ such that $stt^{\prime}=tst^{\prime\prime}$. Then $t^{\prime}t^{\prime\prime}=tsts\neq e$. By Lemma 2.4 we therefore have a dihedral subgroup $W^{\prime}=\langle sts,t,t^{\prime},t^{\prime\prime}\rangle$ of $W$, and $W^{\prime}$ is simply laced since $W$ is (by Lemma 2.6). Clearly $D\subseteq W^{\prime}$. Since $W^{\prime}$ has no more than six elements, $W^{\prime}=D$. So $stt^{\prime}=x\in D$. Since there is a directed edge from $st$ to $x$, $x=t\nleq v$. Since $(st,sts,ts)$ is a broken rhombus of $[e,v]$, by Corollary 2.15, $X_{v}$ is not smooth. ∎

When $W$ is not simply laced, there may exist an involution $w\in W$ which is not the longest element of any parabolic subgroup of $W$ but for which $X_{w}$ is smooth. For example, in type $C_{2}$ there are two involutions of length three. One of them indexes a smooth Schubert variety (and, as was mentioned above, the other one indexes a rationally smooth but not smooth Schubert variety). This example shows that Theorem 3.1 cannot be extended to multiply laced types. We do, however, not know what happens in infinite simply laced types.