Bigrassmannian permutations and Verma modules

Abstract

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $w\in S_n$ into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

1 Introduction, motivation and description of results

1.1 Setup

Consider the symmetric group $S_n$ on $\{1,2,\dots ,n\}$ as a Coxeter group with simple reflections $s_{i}$, given by the elementary transpositions $(i,i+1)$, where $i=1,2,\dots ,n-1$. Denote by $\le$ the corresponding Bruhat order. Given $w\in S_n$, we call i a left descent of w if $s_iw<w$, and a right descent of w if $ws_i<w$. An element $w\in S_n$ is called bigrassmannian provided that it has exactly one left descent, and exactly one right descent. Various aspects of bigrassmannian permutations were studied in [12, 13, 22, 23, 25, 30, 31]. The most relevant for this paper is the property that bigrassmannian permutations are exactly the join-irreducible elements of $S_n$ with respect to $\le$, see [25]. Join-irreducible elements of Weyl groups appear in representation-theoretic context in [18]. Bigrassmannian elements are the first protagonists of the present paper. We denote by ${\mathbf {B}}_n$ the set of all bigrassmannian permutations in $S_n$.

Denote by $\mathfrak {g}$ the simple Lie algebra $\mathfrak {sl}_n$ over $\mathbb {C}$ with the standard triangular decomposition

$$\begin{aligned} \mathfrak {g}=\mathfrak {n}_-\oplus \mathfrak {h}\oplus \mathfrak {n}_+, \end{aligned}$$

where $\mathfrak {h}$ is the Cartan subalgebra of all (traceless) diagonal matrices and $\mathfrak {n}_+$ is the nilpotent subalgebra of all strictly upper triangular matrices. Consider the BGG category $\mathcal {O}$ associated to this triangular decomposition and its principal block $\mathcal {O}_0$, see [7, 16].

The group $S_n$ is the Weyl group of $\mathfrak {g}$, and thus acts naturally on $\mathfrak {h}^*$. We also consider the dot-action of $S_n$ on $\mathfrak {h}^*$, that is

$$\begin{aligned} w\cdot \lambda = w(\lambda +\rho )-\rho \end{aligned}$$

for $\lambda \in \mathfrak {h}^*$, where $\rho$ is the half of the sum of all positive roots in $\mathfrak {h}^*$. For $\lambda \in \mathfrak {h}^*$, we denote by $\Delta (\lambda )$ the Verma module with highest weight $\lambda$ and by $L(\lambda )$ the unique simple quotient of $\Delta (\lambda )$, see [10, 16, 38]. The isomorphism classes of simple objects in $\mathcal {O}_0$ are naturally indexed by the elements in $S_n$ as follows: $S_n\ni w\mapsto L(w\cdot 0)$, where 0 denotes the zero element of $\mathfrak {h}^*$. For $w\in S_n$, we set $\Delta _w:=\Delta (w\cdot 0)$ and $L_w:=L(w\cdot 0)$.

It is well-known, see [10, Chapter 7], that, for $x,y\in S_n$, we have:

• $\dim \mathrm {Hom}_{\mathfrak {g}}(\Delta _x,\Delta _y)\le 1$,

• a non-zero homomorphism from $\Delta _x$ to $\Delta _y$ exists if and only if $x\ge y$,

• each non-zero homomorphism from $\Delta _x$ to $\Delta _y$ is injective.

In particular, all $\Delta _w$, where $w\in S_n$, are uniquely determined submodules of the dominant Verma module $\Delta _e$.

The composition multiplicity of $L_x$ in $\Delta _x$ can be computed in terms of Kazhdan–Lusztig (KL) polynomials, by [2, 8, 11, 19], see Sect. 2.3. The associated KL combinatorics divides $S_n$ into subsets, called two-sided cells, ordered with respect to the two-sided order $\le _{\mathtt {J}}$. This coincides with the division of $S_n$ given by the Robinson-Schensted correspondence: to each $w\in S_n$, we associate a pair $(P_w,Q_w)$ of standard Young tableaux of the same shape $\lambda$ which is a partition of n, see [32,33,34]. The two-sided cells in $S_n$ are indexed by such $\lambda$ and correspond precisely to the fibers of the map from $S_n$ to the set of all partitions of n induced by the Robinson–Schensted correspondence. Moreover, the two-sided order coincides with the dominance order on partitions, see [15]. The longest element $w_0$ of $S_n$ forms a two-sided cell which is the maximum with respect to the two-sided order. If we delete this two-sided cell, in what is left there is again a unique maximum two-sided cell, which we denote by $\mathcal {J}$ and call the penultimate two-sided cell. The two-sided cell ${\mathcal {J}}$ is the second protagonist of the present paper. Under the Robinson-Schensted correspondence, it corresponds to the partition $(2,1^{n-2})$ of n. Under the involution $x\mapsto xw_0$, the two-sided cell $\mathcal {J}$ corresponds to the two-sided cell which contains all simple reflections, studied in, for example, [20, 21]. The Kazhdan–Lusztig cell representation of $S_n$ associated with (any left cell inside) $\mathcal {J}$ is exactly the representation of $S_n$ on $\mathfrak {h}^*$. For each $i,j\in \{1,2,\dots ,n-1\}$, the cell $\mathcal {J}$ contains a unique element w such that $s_iw>w$ and $ws_j>w$. We denote this w by $w_{i,j}$. The image of $w_{i,j}$ under the involution $x\mapsto xw_0$ is the unique element ${\tilde{w}}_{i,j}$ of $S_n$ with left decent $s_i$ and right descent $w_0s_jw_0$. The element ${\tilde{w}}_{i,j}$ is just the product of simple reflections along the unique path in the Dynkin diagram from the vertex corresponding to $s_i$ to the vertex corresponding to $w_0s_jw_0$, written from left to right.

1.2 Motivation

The original motivation for this paper comes from a question by Sascha Orlik and Matthias Strauch which is discussed in more detail in the next subsection. A very special case of that question leads to the problem of determining $\mathrm {Ext}_{{\mathcal {O}}}^1(L_x,\Delta _y)$ in the case $x\ne w_0$. In the case $x=w_0$, we note that $L_{w_0}=\Delta _{w_0}$ and hence the computation of $\mathrm {Ext}_{{\mathcal {O}}}^1(L_x,\Delta _y)$ can be reduced, using twisting functors, to [28, Theorem 32] (see the proof of Corollary 2). The case $x\ne w_0$ requires new techniques and is completed in Corollary 2.

1.3 Description of the main results

The main result of this paper is the following theorem which, in particular, reveals a completely unexpected connection between ${\mathbf {B}}_n$ and $\mathcal {J}$.

Theorem 1

• (i) For $w\in S_n$, the module $\Delta _e/\Delta _w$ has simple socle if and only if $w\in {\mathbf {B}}_n$.

• (ii) The map ${\mathbf {B}}_n\ni w\mapsto \mathrm {soc}(\Delta _e/\Delta _w)$ induces a bijection between ${\mathbf {B}}_n$ and simple subquotients of $\Delta _e$ of the form $L_x$, where $x\in \mathcal {J}$.

• (iii) For $w\in S_n$, the simple subquotients of $\Delta _e/\Delta _w$ of the form $L_x$, where $x\in \mathcal {J}$, correspond under the bijection from (ii), to $y\in {\mathbf {B}}_n$ such that $y\le w$.

• (iv) For $w\in S_n$, the socle of $\Delta _e/\Delta _w$ consists of all $L_x$, where $x\in \mathcal {J}$, which correspond, under the bijection from (ii), to the Bruhat maximal elements in $\{y\in {\mathbf {B}}_n\,:\, y\le w\}$.

The bijection from Theorem 1(ii) is explicitly given in Sect. 4.2. One of the ways to think about this bijection is as follows: each graded simple module $\mathrm {soc}(\Delta _e/\Delta _w)$, where $w\in {\mathbf {B}}_n$, has multiplicity one in $\Delta _e$ (see Proposition 12), moreover, the images of these $\mathrm {soc}(\Delta _e/\Delta _w)$ in the graded Grothendieck group of $\mathcal {O}_0$ are linearly independent.

Theorem 1 provides a categorical, or, alternatively, a representation theoretic interpretation of the poset ${\mathbf {B}}_n$. We note that this interpretation is completely different from the one in [18]. The crucial step towards the formulation of Theorem 1 was an accidental numerical observation which can be found in Corollary 13.

For $x\in S_n$, denote by $\ell (x)$ the length of x (as an element of a Coxeter group), and by ${\mathbf {c}}(x)$, the content of x, that is the number of different simple reflections appearing in a reduced expression of x (this number does not depend on the choice of a reduced expression). Denote by $\Phi :{\mathbf {B}}_n\rightarrow \mathcal {J}$ the map given by $\Phi (w)=w_{i,j}$, if $w\in {\mathbf {B}}_n$ is such that $s_iw<w$ and $ws_j<w$. For $x\in S_n$, denote by $\mathbf {BM}(x)$ the set of all Bruhat maximal elements in the set ${\mathbf {B}}(x):=\{y\in {\mathbf {B}}_n\,:\,y\le x \}$. For $w\in S_n$, we denote by $\nabla _x$ the dual Verma module obtained from $\Delta _x$ by applying the simple preserving duality on $\mathcal {O}_0$, see [16].

Theorem 1, combined with [28, Theorem 32], has the following homological consequence:

Corollary 2

Let $x,y\in S_n$. Then we have

$$\begin{aligned} \dim \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( L_x,\Delta _y\right) =\dim \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( \nabla _y,L_x\right) = {\left\{ \begin{array}{ll} {\mathbf {c}}(xy), &{} x=w_0;\\ 1, &{} x\in \Phi (\mathbf {BM}(y));\\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Theorem 1 and Corollary 2 are proved in Sect. 2. Both Theorem 1 and Corollary 2 extend to singular blocks of ${\mathcal {O}}$, which we prove in Theorem 16 and Proposition 15 in Sect. 3.

The starting point of this paper was a question the second author received from Sascha Orlik and Matthias Strauch, namely, whether, for $x,y\in S_n$ such that $x<y$, the space $\mathrm {Ext}_{{\mathcal {O}}}^1(\Delta _y/\Delta _{w_0},\Delta _x)$ can be non-zero?

From Corollary 2, it follows that the answer is “yes”. For example, for the algebra $\mathfrak {sl}_4$, one can take $y=s_2w_0$, in which case $\Delta _y/\Delta _{w_0}\cong L_y$, and $x=s_2$. By Corollary 2, we have $\mathrm {Ext}_{{\mathcal {O}}}^1(L_y,\Delta _x)=\mathbb {C}$.

1.4 Description of additional results

In Sect. 4, we relate the main results with a number of combinatorial tools. In Sect. 4.1 we explain how the combinatorics and the Hasse diagram of the poset ${\mathbf {B}}_n$ appear naturally in our representation theoretic interpretation of this poset. In Sect. 4.3 we establish representation theoretic significance of the notions of the essential set of a permutation and the associated rank function. In particular, in Corollary 19 we show that the socular constituents in the cokernel of the inclusion of a Verma module indexed by $w\in S_n$ into the dominant Verma module are determined by the essential set of w. Further, in Proposition 22 we show how the associated rank function can be used to compute the degrees of these simple socular constituents in the graded picture.

2 Proofs of the main results

2.1 Category $\mathcal {O}$ tools

Denote by $\underline{\mathcal {O}}_0$ the full subcategory of ${\mathcal {O}}_0$ which consists of all modules on which the center $Z(\mathfrak {g})$ of the universal enveloping algebra $U(\mathfrak {g})$ acts diagonalizably. Note that all Verma modules and all simple modules in ${\mathcal {O}}_0$ are objects in $\underline{\mathcal {O}}_0$. By [35], the category $\underline{\mathcal {O}}_0$ has an auto-equivalence, denoted by $\Theta$, which maps $L_x$ to $L_{x^{-1}}$, for $x\in S_n$. Consequently, $\Theta (\Delta _x)\cong \Delta _{x^{-1}}$, for $x\in S_n$. We note that $\Theta$ does not extend to the whole of ${\mathcal {O}}_0$.

For $x\in S_n$, we have the corresponding twisting functor $T_x$ on $\mathcal {O}_0$, see [1], and the corresponding shuffling functor $C_x$ on $\mathcal {O}_0$, see [9, 29]. Both $\mathcal {L}T_x$ and $\mathcal {L}C_x$ are self-equivalences of the bounded derived category $\mathcal {D}^b(\mathcal {O})$. In the proof below we usually use twisting functors and $\Theta$, however, one can, alternatively, use twisting and shuffling functors or shuffling functors and $\Theta$.

The category $\mathcal {O}_0$ is equivalent to the category of finite dimensional modules over a finite dimensional basic algebra, which we denote by A. The algebra A is unique, up to isomorphism. By [36], it is Koszul and hence admits a Koszul $\mathbb {Z}$-grading. We denote by ${}^{\mathbb {Z}}\mathcal {O}_0$ the category of finite dimensional $\mathbb {Z}$-graded A-modules. We denote by $\langle k\rangle$ the grading shift on ${}^{\mathbb {Z}}\mathcal {O}_0$ normalized such that it maps degree k to degree zero, and we fix the standard graded lifts of $L_w$ concentrated in degree zero, and of $\Delta _w$ such that its top is concentrated in degree zero.

2.2 Potential socle of the cokernel of an inclusion of Verma modules

Proposition 3

Let $x,y,z\in S_n$ be such that $x\ge y$ and $L_z$ is in the socle of $\Delta _y/\Delta _x$. Then $z\in \mathcal {J}$.

Proof

As $\Delta _y/\Delta _x$ is a submodule of $\Delta _e/\Delta _x$, it is enough to prove the claim in the case $y=e$.

Consider first the case $x=w_0$. Then $\Delta _{w_0}$ is the socle of $\Delta _e$ and we claim that the socle of $\Delta _e/\Delta _{w_0}$ consists of all $L_{x}$, where $x\in W$ is such that $\ell (x)=\ell (w_0)-1$. The best way to argue that the socle of $\Delta _e/\Delta _{w_0}$ is as described above is to recall that $\Delta _e$ is rigid, see [3, 17]. This means that the socle filtration and the radical filtration of $\Delta _e$ coincide. Moreover, these two filtrations also coincide with the graded filtration, see [3, Proposition 2.4.1]. The submodule $\Delta _{w_0}=L_{w_0}$ in $\Delta _e$ lives in degree $\ell (w_0)$. By rigidity, the socle of $\Delta _e/\Delta _{w_0}$ consists of simple modules which live in degree $\ell (w_0)-1$. Since, by KL combinatorics, a simple module $L_x$ appears in $\Delta _e$ only in degrees $\le \ell (x)$ and $L_{w_0}$ has multiplicity one, the only simples in degree $\ell (w_0)-1$ are those $L_x$ for which $\ell (x)=\ell (w_0)-1$. Note also that all such x belong to $\mathcal {J}$. This proves the claim of the proposition in the case $x=w_0$.

For $x\ne w_0$, using the self-equivalence $\mathscr {L}T_{w_0x^{-1}}$, we have

$$\begin{aligned} \mathrm {Hom}_{\mathcal {D}^b(\mathcal {O})}\left( L_z,\Delta _e/\Delta _x\right) \cong \mathrm {Hom}_{\mathcal {D}^b(\mathcal {O})}\left( \mathscr {L}T_{w_0x^{-1}}(L_z), \Delta _{w_0x^{-1}}/\Delta _{w_0}\right) . \end{aligned}$$

(1)

By the previous two paragraphs, the socle of $\Delta _{w_0x^{-1}}/\Delta _{w_0}$ consist of the simples $L_w$, where $w\in \mathcal {J}$. Therefore, for the right hand side of (1) to be non-zero, the module $T_{w_0x^{-1}}(L_z)$ must contain some simple subquotient of the form $L_w$, where $w\in \mathcal {J}$. From [1, Theorem 6.3], it follows that all simple subquotients of $T_{w_0x^{-1}}(L_z)$ are of the form $L_u$, where $u\le _{\mathtt {J}}z$. This yields $z\in \mathcal {J}$, completing the proof. $\square$

Proposition 4

Let $x\in S_n$ and s be a simple reflection.

• (i) If $sx<x$, then the socle of $\Delta _e/\Delta _x$ contains some $L_y$ such that $sy>y$.

• (ii) If $xs<x$, then the socle of $\Delta _e/\Delta _x$ contains some $L_y$ such that $ys>y$.

Proof

Due to the assumption $sx<x$, we have $\Delta _x\subset \Delta _{sx}$, in particular, the socle of $\Delta _e/\Delta _x$ contains the socle of $\Delta _{sx}/\Delta _x$. The module $\Delta _{sx}/\Delta _x$ is non-zero and, by construction, s-finite (i.e. the action on this module of the $\mathfrak {sl}_2$-subalgebra of $\mathfrak {g}$ corresponding to s is locally finite). In particular, any $L_y$ in the socle of $\Delta _{sx}/\Delta _x$ is also s-finite. Therefore $sy>y$ for each y such that $L_y$ is in the socle of $\Delta _{sx}/\Delta _x$. Claim (i) follows. Claim (ii) follows from claim (i) using $\Theta$. $\square$

Corollary 5

Let $w\in S_n$ be such that the socle of $\Delta _e/\Delta _w$ is simple. Then w is a bigrassmannian permutation.

Proof

Assume that s and t are different simple reflections such that $sw<w$ and $tw<w$. By Proposition 4, the socle of $\Delta _e/\Delta _w$ contains some $L_y$ such that $sy>y$ and some $L_z$ such that $tz>z$. Both $y,z\in \mathcal {J}$. Note that, for each element u of $\mathcal {J}$, there is at most one simple reflection r such that $ru>u$. This means that $y\ne z$ and hence the socle of $\Delta _e/\Delta _w$ is not simple. A similar argument works in the case when there are different simple reflections s and t such that $ws<w$ and $wt<w$. The claim follows. $\square$

Proposition 6

Assume that $x\in S_n$ and $y\in \mathcal {J}$ be such that $L_y$ is in the socle of $\Delta _e/\Delta _x$. Assume that s is a simple reflection such that $sy>y$. Then $sx<x$.

Proof

Assume $sx>x$. Applying $\mathscr {L}T_s$ and using [1, Theorem 6.1], similarly to (1), we have

$$\begin{aligned} \mathrm {Hom}_{\mathcal {D}^b(\mathcal {O})}\left( L_y,\Delta _e/\Delta _x\right) \cong \mathrm {Hom}_{\mathcal {D}^b(\mathcal {O})}\left( L_z[1], \Delta _{s}/\Delta _{sx}\right) . \end{aligned}$$

The right hand side of this equality is 0 since $\Delta _{s}/\Delta _{sx}$ is a module in homological position 0 while $L_z[1]$ is a module in homological position $-1$. The claim follows.

$\square$

2.3 Combinatorial tools

Consider the Hecke algebra $\mathbb {H}_n$ associated to the Coxeter system $(S_n,S)$, where S is the set $\{s_1,\ldots , s_{n-1}\}$ of simple reflections. $\mathbb {H}_n$ is a ${\mathbb {Z}}[v,v^{-1}]$-algebra generated by $H_i$, for $1\le i\le n-1$, which satisfy the braid relations

$$\begin{aligned} H_iH_{i+1}H_i=H_{i+1}H_iH_{i+1}, \end{aligned}$$

for all $1\le i\le n-2$, and the quadratic relation

$$\begin{aligned} \left( H_i+v\right) \left( H_i-v^{-1}\right) =0, \end{aligned}$$

for all $1\le i\le n-1$. Given a reduced expression $w=s_i s_j\cdots s_k$ of $w\in S_n$, we let $H_w=H_iH_j\cdots H_k$. The element $H_w$ is independent of the choice of the reduced expression, and $\{H_w\,:\,w\in S_n\}$ is a ($\mathbb Z[v,v^{-1}]$-)basis of ${\mathbb {H}}_n$ called the standard basis. Consider the (${\mathbb {Z}}$-algebra-)involution on $H_n$ denoted by a bar, such that ${\overline{v}}= v^{-1}$ and $\overline{H_i} = H_i^{-1}$. Then there is a unique element ${\underline{H}}_w$ in ${\mathbb {H}}_n$ such that $\overline{{\underline{H}}_w} = {\underline{H}}_w$ and

$$\begin{aligned} {\underline{H}}_w = H_w + \sum _y p_{y,w}H_y, \end{aligned}$$

for some $p_{y,w}\in v{\mathbb {Z}}[v]$. The elements ${\underline{H}}_w$, where $w\in W$, form a basis of ${\mathbb {H}}_n$ called the Kazhdan–Lusztig (KL) basis. Given $w,y\in S_n$, the coefficient of v in $p_{w,y}+p_{y,w}$ is denoted by $\mu (w,y)=\mu (y,w)$, defining the (Kazhdan–Lusztig), $\mu$-function. If $s\in S$ then ${\underline{H}}_s=H_s+v$ and we have

$$\begin{aligned} {\underline{H}}_s{\underline{H}}_y = (v+v^{-1}){\underline{H}}_{y} , \end{aligned}$$

(2)

for $sy<y$, and

$$\begin{aligned} {\underline{H}}_s{\underline{H}}_y = {\underline{H}}_{sy}+\sum _{sx<x, x<y}\mu (x,y){\underline{H}}_x, \end{aligned}$$

(3)

for $sy>y$. Another basic fact is that

$$\begin{aligned} p_{x,w_0}=v^{\ell (w_0)-\ell (x)}. \end{aligned}$$

(4)

For more details about KL basis we refer to [19].

A consequence of the Kazhdan–Lusztig conjecture (which is now a theorem, see [2, 8, 11, 19]), is that the polynomial $p_{x,y}$ above describes the composition multiplicities of the Verma modules in $^{\mathbb {Z}}{\mathcal {O}}_0$ in the following sense:

$$\begin{aligned} \sum _{i\in \mathbb {Z}} [\Delta _w:L_y\langle - i\rangle ]v^i = p_{w,y}. \end{aligned}$$

From the Kazhdan–Lusztig conjecture/theorem, combined with the Koszul self-duality of ${\mathcal {O}}_0$ from [3, 36], it follows that the integer $\mu (w,y)$ equals the dimension of the first extension space between $L_w$ and $L_y$ (and hence determines the Gabriel quiver of the algebra A).

We denote by ${\mathbf {a}} :W \rightarrow \mathbb {Z}_{\ge 0}$ Lusztig’s ${\mathbf {a}}$-function, see [27]. The value ${\mathbf {a}}(w)$ does not change when w varies over a two-sided cell. We write ${\mathbf {a}}({\mathcal {J}})={\mathbf {a}}(w)$ for $w\in {\mathcal {J}}$. By definition, the value ${\mathbf {a}}(w)$ for $w \in \mathcal {J}$ describes the minimal possible degree shift for simple subquotients $L_u$, with $u\in \mathcal {J}$, of $\Delta _e$ (i.e. the degree shift for the top layer of the tetrahedron on Fig. 2). The minimal shift is achieved exactly when w is a Duflo element. It follows that

$$\begin{aligned} p_{e,w}\in {\mathbb {Z}} \{ v^a\ |\ {\mathbf {a}}(w)\le a\le \ell (w)\} \end{aligned}$$

(5)

where the first equality holds if and only if w is a Duflo element. Since $\mathcal {J}$ contains $w_{1,1}$, which is the longest element of some parabolic subgroup of $S_n$, we have ${\mathbf {a}}(\mathcal {J})=\ell (w_{1,1})=\frac{(n-1)(n-2)}{2}$.

Proposition 7

Let $x,y\in {\mathcal {J}}$. Then $\mu (x,y)\ne 0$ if and only if x, y are adjacent in the Bruhat graph. In the latter case, we have $\mu (x,y)=1$.

Proof

By the Kazhdan–Lusztig inversion formula, see [19], (or, equivalently, by Koszul duality, see [3]), we have $\mu (x,y)=\mu (w_0x^{-1},w_0y^{-1})$. Now, the claim follows from the similar fact about the small two-sided cell (see [26, Proposition 3.8 (d)] or [20, 21]). $\square$

For convenience of the reader, on Fig. 1 we give the Bruhat graph of $\mathcal {J}$ which is also illustrated by the explicit special for $n=2,3,4$.

Fig. 1

Bruhat graph of the penultimate cell (the rows are right, and the columns are left cells) with the explicit special cases $n=3,4$

Lemma 8

Suppose $y=w_{i,j}\in {\mathcal {J}}$, with $i\ne j$. Then

$$\begin{aligned} p_{s,y} = v^{-1}p_{e,y},\quad \text { for all } s \in S. \end{aligned}$$

(6)

Proof

Suppose $s\ne s_i$. Then, comparing the $H_e$-coefficients in (2), gives (6). If $s=s_i$, then we have $s\ne s_j$, so comparing the $H_e$-coefficients in the equation ${\underline{H}}_y{\underline{H}}_s=(v+v^{-1}){\underline{H}}_y$ gives (6). $\square$

Lemma 9

Suppose $y=w_{i,j}\in {\mathcal {J}}$, with $i\ne j$, $s\in S$ and $sy>y$, we have

$$\begin{aligned} (v+v^{-1})p_{e,y}=p_{e,sy} + \sum _{\begin{array}{c} u \in S \\ uy\sim _{\mathtt {L}} y \\ uy<y \end{array}} p_{e,uy}. \end{aligned}$$

(7)

Proof

Consider Equation (3), for $y\in {\mathcal {J}}$. Since all x appearing in this equation with non-zero coefficient satisfy, at the same time, $x<y$ and $x\ge _{\mathtt {L}} y$, and, since $y\in {\mathcal {J}}$, we have $x\sim _{\mathtt {L}} y$. Now, comparing the $H_e$-coefficients on both sides in Equation (3), we get

$$\begin{aligned} (v+v^{-1})p_{e,y}=p_{s,y} + v p_{e,y}=p_{e,sy} + \sum _{\begin{array}{c} x\sim _{\mathtt {L}} y \\ x<y \end{array}} \mu (x,y)p_{e,x}, \end{aligned}$$

(8)

where we used (6) for the first equality. Then Proposition 7 reduces (8) to (7). $\square$

Proposition 10

Let $y=w_{1,1}$ or $y=w_{n-1,n-1}$. Then $p_{e,y}=v^{\ell (y)}$.

Proof

The element $w_{n-1,n-1}$ is the longest element for the parabolic subgroup of $S_n$ generated by $\{1,\ldots , n-2\}$. The element $w_{1,1}$ is the longest element for the parabolic subgroup of $S_n$ generated by $\{2,\ldots ,n-1\}$. Since the KL basis can be computed in the parabolic subgroups, the claim of the lemma follows from (4) applied to the Coxeter group $S_{n-1}$ with the corresponding identification of simple reflections. $\square$

For $i,j \in \{ 0, \ldots ,n\}$, put $p_{i,j} := p_{e,w_{i,j}}$ if $1\le i,j \le n-1$, and $p_{i,j} :=0$ otherwise.

Lemma 11

Suppose $w_{i,j}\in {\mathcal {J}}$, with $i\ne j$. Then

$$\begin{aligned} \left( v+v^{-1}\right) p_{i,j} = p_{i-1,j} + p_{i+1,j} + \delta _{i,n-j} \cdot v^{\ell (w_0)}, \end{aligned}$$

(9)

where $\delta$ denotes the Kronecker symbol.

Proof

This follows from (7) with $s=s_i$, (4) and Fig. 1. $\square$

Proposition 12

Let $w=w_{i,j}\in {\mathcal {J}}$. We have

$$\begin{aligned} p_{e,w}=v^{\ell (w)}+v^{\ell (w)-2}+\cdots +v^{\ell (w)-2d(w)}, \end{aligned}$$

(10)

where

$$\begin{aligned} d(w)={\text {min}}\{i-1,j-1, n-1-i,n-1-j\}. \end{aligned}$$

Proof

We proceed by induction on $d=d(w)$.

Let $d=0$. By symmetry (flipping the Dynkin diagram on one hand, and taking inverses of permutations on the other hand), it is enough to consider $w=w_{i,1}$. If $i=1$, then Proposition 10 gives (10). If $i=2$, then (9) gives

$$\begin{aligned} (v+v^{-1})p_{2,1}= v^{a} + p_{3,1}, \end{aligned}$$

where $a:={\mathbf {a}}(\mathcal {J})$. Since $\ell (w_{2,1})=a+1$, by (5), we have $p_{2,1}=c v^{a+1}$ for some $c \in {\mathbb {Z}}$. For the same reason, $p_{3,1}$ does not have term proportional to $v^a$, so we get $p_{2,1} = v^{a+1}$ and $p_{3,1} = v^{a+2}$. From this we can obtain (10) for any $w_{i,1}$, using (9) and a two-step induction.

Let $d = d(w)>0$. By symmetry, we may assume $w=w_{d+1,j}$, with $d<j<n-d$. We assume that (10) is true for $w_{d,1}$ and (if exists) for $w_{d-1,1}$. Equation (9) applied to $w=w_{d,j}$ gives

$$\begin{aligned} \left( v+v^{-1}\right) p_{d,j} = p_{d-1,j} + p_{d+1,j}. \end{aligned}$$

From this, it easily follows that (10) is also true for $w_{d+1,j}$. $\square$

For $i,j\in \{1,2,\dots ,n-1\}$, we denote by ${\mathbf {B}}_n^{(i,j)}$ the set of all $w\in {\mathbf {B}}_n$ such that $s_iw<w$ and $ws_j<w$.

Corollary 13

We have

$$\begin{aligned} |{\mathbf {B}}_n|=\sum _{w\in \mathcal {J}}p_{e,w}(1), \end{aligned}$$

(11)

moreover, for each $i,j\in \{1,2,\dots ,n-1\}$, we have $|{\mathbf {B}}_n^{(i,j)}|=p_{e,w_{i,j}}(1)$.

Proof

By Proposition 12, the right hand side of (11) evaluates to the tetrahedral number $\frac{(n-1)n(n+1)}{6}$. That the left hand side of (11) is given by the same number follows from the main result of [23]. The refined version, for each $i,j\in \{1,2,\dots ,n-1\}$, follows from the main result of [12]. Both statements are best seen by comparing the main result of [12] to the picture described in Sect. 4.1, where the right hand side (11) is realized as a tetrahedron in $\mathbb {R}^3$ and the elements corresponding to ${\mathbf {B}}_n^{(i,j)}$ consist of all elements of this tetrahedron aligned along a vertical line. $\square$

Proposition 14

Let $i,j\in \{1,2,\dots ,n-1\}$. The restriction of the Bruhat order to each ${\mathbf {B}}_n^{(i,j)}$ is a chain, moreover, for $x,y\in {\mathbf {B}}_n^{(i,j)}$ such that $x<y$, there exist $w\in {\mathbf {B}}_n{\setminus }{\mathbf {B}}_n^{(i,j)}$ such that $x<w<y$.

Proof

This follows directly from the main result of [12]. The poset ${\mathbf {B}}_n$ can be realized as a tetrahedron in $\mathbb {R}^3$, see Sect. 4.1. The subset ${\mathbf {B}}_n^{(i,j)}$ consist of all elements of this tetrahedron aligned along a vertical line. If such a line contains more than one element of the tetrahedron, we have a diamond shaped subset of ${\mathbf {B}}_n$ and the elements of this diamond outside the original line provide the necessary w, see for example the points (2, 2, 5) and (2, 2, 3), in the case $n=4$, where w can be chosen as any of the elements (2, 1, 4), (1, 2, 4), (3, 2, 4) or (2, 3, 4). $\square$

2.4 Proof of Theorem 1(i), (ii)

Corollary 5 gives one direction of (i). For the other direction, we need that, for any $w\in {\mathbf {B}}_n$, the module $\Delta _e/\Delta _w$ has simple socle. Assume that $s_i$ and $s_j$ are such that $s_iw<w$ and $ws_j<w$. Then, by Proposition 6, $L_{w_{i,j}}$ is the only possible simple subquotient in the socle of $\Delta _e/\Delta _w$. We need to prove that the multiplicity of $L_{w_{i,j}}$ in the socle of $\Delta _e/\Delta _w$ equals 1.

By Proposition 14, there is a chain

$$\begin{aligned} w_1<u_1<w_2<u_2<\dots<w_{k-1}<u_{k-1}<w_k, \end{aligned}$$

where ${\mathbf {B}}_n^{(i,j)}=\{w_1,w_2,\dots ,w_k\}$ and all $u_l \in {\mathbf {B}}_n{\setminus } {\mathbf {B}}_n^{(i,j)}$. This chain gives an a sequence of inclusions

$$\begin{aligned} \Delta _e\supsetneq \Delta _{w_1}\supsetneq \Delta _{u_1}\supsetneq \Delta _{w_2}\supsetneq \Delta _{u_2} \supsetneq \dots \supsetneq \Delta _{w_k} \end{aligned}$$

which, in turn, gives rise to a sequence of projections

$$\begin{aligned} \Delta _e/\Delta _{w_1}\twoheadleftarrow \Delta _e/\Delta _{u_1}\twoheadleftarrow \Delta _e/\Delta _{w_2}\twoheadleftarrow \Delta _e/\Delta _{u_2}\twoheadleftarrow \dots \twoheadleftarrow \Delta _e/\Delta _{w_k}. \end{aligned}$$

This implies that the socle of each $\Delta _e/{\Delta _{w_l}}$ contains a summand which is not a summand of the socle of $\Delta _e/{\Delta _{w_m}}$, for any ${m<l}$. From Corollary 13 we obtain that $k={ p}_{e,w_{i,j}}(1)$ and hence the socle of each $\Delta _e/{\Delta _{w_l}}$ contains a unique summand which is not in the socle of any $\Delta _e/{\Delta _{w_m}}$, for ${m<l}$.

Therefore, for both claims (i) and (ii), it is enough to prove that the socle of each $\Delta _e/{\Delta _{w_l}}$ is simple. We argue that the socle constituents of any $\Delta _e/{\Delta _{w_m}}$, where ${m<l}$, cannot contribute to the socle of $\Delta _e/{\Delta _{w_l}}$. Assume that this is not the case and some socle constituent of some $\Delta _e/{\Delta _{w_m}}$ contributes to the socle of $\Delta _e/{\Delta _{w_l}}$. Then it also must contribute to the socle of $\Delta _e/{\Delta _{u_{l-1}}}$, since ${w_m<u_{l-1}<w_l}$. But this contradicts Proposition 6 and completes the proof.

2.5 Proof of Theorem 1(iii) and (iv)

Both statements follow from the bijection given in Theorem 1(ii) and the fact that $\Delta _x\subset \Delta _y$ is equivalent to $x\ge y$.

2.6 Proof of Corollary 2

First of all, we note that the equality

$$\begin{aligned} \dim \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( L_x,\Delta _y\right) =\dim \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( \nabla _y,L_x\right) \end{aligned}$$

follows by applying the simple preserving duality on $\mathcal {O}_0$. Therefore we only need to prove the following:

$$\begin{aligned} \dim \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( L_x,\Delta _y\right) = {\left\{ \begin{array}{ll} {\mathbf {c}}(xy), &{} x=w_0;\\ 1, &{} x\in \Phi (\mathbf {BM}(y));\\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(12)

Consider first the case $x=w_0$. In this case $L_{w_0}=\Delta _{w_0}$. Applying the twisting functor $T_{y}$ and using the fact that it is acyclic on Verma modules, see [1, Theorem 2.2], we observe that

$$\begin{aligned} \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( \Delta _{y^{-1}w_0},\Delta _e\right) = \mathrm {Ext}_{{\mathcal {O}}}^{1}\left( \Delta _{w_0},\Delta _y\right) . \end{aligned}$$

By [28, Theorem 32], the dimension of $\mathrm {Ext}_{{\mathcal {O}}}^{1}(\Delta _{y^{-1}w_0},\Delta _e)$ equals ${\mathbf {c}}(y^{-1}w_0)={\mathbf {c}}(xy)$. This establishes (12) in the case $x=w_0$.

Assume now that $x\ne w_0$. Let

$$\begin{aligned} 0\rightarrow \Delta _y\rightarrow M \rightarrow L_x \rightarrow 0 \end{aligned}$$

(13)

be a non-split short exact sequence. Since $L_x$ is a simple object and (13) is non-split, the socle of M coincides with the socle of $\Delta _y$ and hence is isomorphic to $L_{w_0}$. In particular, M is a submodule of the injective envelope $I_{w_0}$ of $L_{w_0}$. As the multiplicity of $L_{w_0}$ in M is one, all nilpotent endomorphisms of $I_{w_0}$ send M to 0. Since $\Delta _e$, being a projective object in ${\mathcal {O}}$, is copresented by projective-injective objects in ${\mathcal {O}}$ (see [24, § 3.1]) and since $I_{w_0}$ is the only indecomposable projective-injective object in ${\mathcal {O}}_0$, it follows that M is a submodule of $\Delta _e$. This means that $L_x\cong M/\Delta _y$ corresponds to a socle constituent of $\Delta _e/\Delta _y$. Recall that, for any module M and any simple module L, the dimension of $\mathrm {Ext}^1(L,M)$ coincides with the multiplicity of L in the socle of I/M, where I is a minimal injective envelope of M. Combining all these observations together, we have that, for $x\ne w_0$, formula (12) follows from Theorem 1(iv).

3 Singular blocks

In this subsection we extend our results on ${\mathcal {O}}_0$ to an arbitrary block ${\mathcal {O}}_\mu$ (that is, the Serre subcategory of ${\mathcal {O}}$ with the simple objects $L(w\cdot \mu )$, for w in the $\mu$-integral subgroup $S_n^{(\mu )}$ of $S_n$) thus to the entire category ${\mathcal {O}}$. Let $\mu \in \mathfrak {h}^*$ be an $S_n^{(\mu )}$-dominant weight. The first, standard, remark is that, up to equivalence with a block for some other n, it is enough to consider the case when $\mu$ is integral and hence $S_n^{(\mu )}=S_n$, see Soergel’s combinatorial description of $\mathcal {O}$ in [36]. Therefore, from now on we assume $\mu$ to be integral.

Note that, if $I= \{i\in \{1,\ldots ,n-1\}\ |\ s_i\cdot \mu =\mu \}$ and $S_I$ is the parabolic subgroup of $S_n$ generated by I, then $S_I\cong S_\mu :={\text {Stab}}_W(\mu )$. So, we have $L(w\cdot \mu )\cong L(wz\cdot \mu )$ and $\Delta (w\cdot \mu )\cong \Delta (wz\cdot \mu )$, for any $z\in S_\mu$. Given $w\in S_n$, we denote by ${\underline{w}}$ and ${\overline{w}}$ the unique shortest and the unique longest coset representatives for the coset $wS_\mu$ in $S_n/S_\mu$. We have the indecomposable projective functors

$$\begin{aligned} \theta ^0_\mu :{\mathcal {O}}_0\rightarrow {\mathcal {O}}_\mu \qquad \text { and }\qquad \theta ^\mu _0:{\mathcal {O}}_\mu \rightarrow {\mathcal {O}}_0 \end{aligned}$$

called translation to the  $\mu$-wall and translation out of the  $\mu$-wall, respectively. (These functors are sometimes denoted by $T_0^\mu$ and $T_\mu ^0$, respectively, e.g., in [16].) They are biadjoint, exact and determined uniquely, up to isomorphism, by $\theta ^0_{\mu }\Delta _e=\Delta _\mu$ and $\theta _0^{\mu }\Delta _\mu =P_{{\overline{e}}}$, respectively, see [5]. In particular, from the biadjointness it follows that the action of the functor $\theta ^0_\mu$ on simple modules is given, for $w\in S_n$, by

$$\begin{aligned} \theta ^0_\mu L_w\cong {\left\{ \begin{array}{ll} L(w\cdot \mu ),&{} w={\overline{w}};\\ 0,&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

On the level of the Grothendieck group, the functor $\theta _0^{\mu }\theta ^0_{\mu }$ corresponds to the right multiplication with the sum of all elements in $S_\mu$, see [5, § 3.4].

Proposition 15

Let $x,y\in S_n$ be such that $x<y$. Then we have

$$\begin{aligned} {\text {soc}}(\Delta (x\cdot \mu )/\Delta (y\cdot \mu )) \cong \theta ^0_\mu \left( {\text {soc}}\Delta _{{\underline{x}}}/\Delta _{{\underline{y}}}\right) . \end{aligned}$$

Proof

We may assume $x={\underline{x}}$ and $y={\underline{y}}$.

Suppose $L_w$ is a socle component of $\Delta _x/\Delta _y$. Then, by exactness, the module $\theta ^0_\mu L_w$, whenever it is nonzero, that is, whenever $w={\overline{w}}$, is a socle component of

$$\begin{aligned} \theta ^0_\mu \left( \Delta _x/\Delta _y\right) \cong \left( \theta ^0_\mu \Delta _x\right) /\left( \theta ^0_\mu \Delta _y\right) \cong \Delta (x\cdot \mu )/\Delta (y\cdot \mu ). \end{aligned}$$

Now, assume that $L(w\cdot \mu )$ is a socle component of $\Delta (x\cdot \mu )/\Delta (y\cdot \mu )$. Then we have

$$\begin{aligned} {\text {Ext}}_{{\mathcal {O}}}^1(L(w\cdot \mu ),\Delta (y\cdot \mu ))\ne 0. \end{aligned}$$

By adjunction, we also have

$$\begin{aligned} {\text {Ext}}_{{\mathcal {O}}}^1(L(w\cdot \mu ),\Delta (y\cdot \mu ))\cong {\text {Ext}}_{{\mathcal {O}}}^1\left( \theta ^0_\mu L_{{\overline{w}}},\Delta (y\cdot \mu )\right) \cong {\text {Ext}}_{{\mathcal {O}}}^1\left( L_{{\overline{w}}},\theta ^\mu _0\Delta (y\cdot \mu )\right) . \end{aligned}$$

As $\theta ^\mu _0$ is a projective functor, $\theta ^\mu _0\Delta (y\cdot \mu )$ has a Verma flag. As, on the level of the Grothendieck group, $\theta _0^{\mu }\theta ^0_{\mu }$ corresponds to the right multiplication with the sum of all elements in $S_\mu$, the Verma flag of $\theta ^\mu _0\Delta (y\cdot \mu )$ has subquotients $\Delta _{yz}$, where $z\in S_\mu$, each appearing with multiplicity one. It follows that ${\text {Ext}}_{{\mathcal {O}}}^1(L_{{\overline{w}}},\Delta _{yz})\ne 0$, for some $z\in S_\mu$, which means ${\overline{w}}\in {\mathcal {J}}$ by Corollary 2. Moreover, we obtain that $L_{{\overline{w}}}$ appears in the socle of $\Delta _e/\Delta _{yz}$. Since $\theta ^0_\mu L_{{\overline{w}}}\ne 0$ while $\theta ^0_\mu (\Delta _{y}/\Delta _{yz})=0$, the subquotient $L_{{\overline{w}}}$ is also contained in the socle of $\Delta _e/\Delta _{y}$. To see that $L_{{\overline{w}}}$ is in the socle of $\Delta _x/\Delta _y$, we note that $\Delta _e/\Delta _x$ cannot contain this subquotient in the socle, for in that case $\theta ^0_\mu (\Delta _e/\Delta _x)\cong \Delta (\mu )/\Delta (x\cdot \mu )$ would contain $L(w\cdot \mu )$ in the socle, contradicting our assumption (note that the graded multiplicities of simples from ${\mathcal {J}}$ in $\Delta _e$, as well as of the corresponding translated simples in $\Delta (\mu )=\theta ^0_\mu \Delta _e$ are one or zero, so we can distinguish all such simple subquotients). The proof is complete. $\square$

Theorem 16

Let $x,y\in S_n$ and let $\mu$ be an integral, dominant weight. Then we have

$$\begin{aligned} \dim \mathrm {Ext}_{{\mathcal {O}}}^{1}(L(x\cdot \mu ),\Delta (y\cdot \mu ))= & {} \dim \mathrm {Ext}_{{\mathcal {O}}}^{1}(\nabla (y\cdot \mu ),L(x\cdot \mu )) \nonumber \\= & {} {\left\{ \begin{array}{ll} {\mathbf {c}}({\overline{x}}{\underline{y}})-|I|, &{} {\overline{x}}=w_0;\\ 1, &{} {\overline{x}}\in \Phi (\mathbf {BM}({\underline{y}}));\\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Proof

Similarly to the proof of Corollary 2, the proof of Proposition 15 takes care of the case ${\overline{x}}\ne w_0$.

For the case ${\overline{x}}=w_0$, we need to generalize [28, Theorem 32], which is proved in the regular setup, to singular blocks. So, let $x=w_0$. We may assume $y={\underline{y}}$. For the proof, we work in the graded category $^{{\mathbb {Z}}}{\mathcal {O}}$, where hom- and ext-functors are, as usual, denoted by $\hom$ and ${\text {ext}}_\mathcal {O}^i$. The precise claim in this case is

$$\begin{aligned} \dim {\text {ext}}_{{\mathcal {O}}}^1\left( L\left( w_0\cdot \mu \right) ,\Delta (y\cdot \mu )\langle i\rangle \right) ={\left\{ \begin{array}{ll}{\mathbf {c}}\left( w_0y\right) -|I|, &{}\text { if}\, i =\ell (w_0y)-2;\\ 0, &{}\text { otherwise.} \end{array}\right. }\qquad \end{aligned}$$

(14)

If $\mu =0$, then, similarly to the proof of Corollary 2, Formula (14) reduces to [28, Theorem 32] using twisting functors.

In the general case, we first note that $L(w_0\cdot \mu )\cong \theta ^0_\mu L_{w_0}$ where we now use the graded translation functor $\theta ^0_\mu :\!^{{\mathbb {Z}}}{\mathcal {O}}_0\rightarrow \!^{{\mathbb {Z}}}{\mathcal {O}}_\mu$, see [37]. By adjunction, this gives

$$\begin{aligned} {\text {ext}}_{{\mathcal {O}}}^1\left( L\left( w_0\cdot \mu \right) ,\Delta (y\cdot \mu )\langle i\rangle \right) \cong {\text {ext}}_{{\mathcal {O}}}^1\left( L_{w_0},\theta ^\mu _0\Delta (y\cdot \mu )\langle i\rangle \right) . \end{aligned}$$

The graded module $\theta ^\mu _0\Delta (y\cdot \mu )$ has a filtration

$$\begin{aligned} 0=M_{-1}\subset M_0\subset M_1\subset \dots \subset M_{\ell ({\overline{e}})}=\theta _\mu ^0\Delta (y\cdot \mu ), \end{aligned}$$

(15)

with subquotients

$$\begin{aligned} M_i/M_{i-1}\cong \bigoplus _{z\in S_\mu :\ell (z)=i} \Delta _{yz}\langle \ell (z) \rangle , \qquad \text { for } i=0,1,2,\dots ,\ell ({\overline{e}}). \end{aligned}$$

In particular, we have the short exact sequence

$$\begin{aligned} 0\rightarrow M_0\rightarrow \theta ^\mu _0\Delta (y\cdot \mu )\rightarrow N\rightarrow 0, \end{aligned}$$

(16)

where $M_0\cong \Delta _y$ and N has an Verma flag induced from (15). Consider the long exact sequence obtained by applying $\hom (L_{w_0},{}_-\langle i\rangle )$ to (16), for $i\in {\mathbb {Z}}$. By (14) applied to $\mu =0$ and $i=\ell (w_0y)-2$, each of the spaces ${\text {ext}}_{{\mathcal {O}}}^1(L_{w_0},\Delta _{yz}\langle \ell (w_0y)-2+\ell (z) \rangle )$ is zero and thus ${\text {ext}}_{{\mathcal {O}}}^1(L_{w_0},N\langle \ell (w_0y)-2 \rangle )=0$. Since $\theta ^\mu _0\Delta (y\cdot \mu )$ has simple socle and this socle lives in degree $\ell (w_0y)$, we also have $\hom (L_{w_0},\theta ^\mu _0\Delta (y\cdot \mu )\langle \ell (w_0y)-2\rangle )=0$. Therefore, our long exact sequence gives the short exact sequence

$$\begin{aligned}&0\rightarrow \hom \left( L_{w_0},N\langle \ell \left( w_0y\right) -2 \rangle \right) \rightarrow {\text {ext}}_{{\mathcal {O}}}^1\left( L_{w_0},\Delta _y\langle \ell \left( w_0y\right) -2 \rangle \right) \\&\quad \rightarrow {\text {ext}}_{{\mathcal {O}}}^1\left( L_{w_0}, \theta ^\mu _0\Delta (y\cdot \mu )\langle \ell (w_0y)-2 \rangle \right) \rightarrow 0. \end{aligned}$$

Using (14), for $\mu =0$, again, the dimension of the middle space in this short exact sequence is given by ${\mathbf {c}}(w_0y)$. Since the socle of $\Delta _{yz}$ in $^\mathbb {Z}{\mathcal {O}}_0$ is $L_{w_0}\langle -\ell (w_0yz) \rangle$, we have

$$\begin{aligned} \hom \left( L_{w_0},N\langle \ell (w_0y)-2 \rangle \right)= & {} \hom \left( L_{w_0},\bigoplus _{s\in I}\Delta _{ys}\langle \ell (s) +\ell \left( w_0y\right) -2\rangle \right) \\= & {} \hom \left( L_{w_0},\bigoplus _{s\in I}\Delta _{ys}\langle \ell (w_0ys)\rangle \right) ={\mathbb {C}}^{|I|}. \end{aligned}$$

It follows that

$$\begin{aligned} \dim {\text {ext}}_{{\mathcal {O}}}^1\left( L_{w_0}, \theta ^\mu _0\Delta (y\cdot \mu )\langle \ell \left( w_0y\right) -2 \rangle \right) ={\mathbf {c}}\left( w_0y\right) -|I|. \end{aligned}$$

It remains to show that, for $i\ne \ell (w_0y)-2$, we have

$$\begin{aligned} {\text {ext}}_{{\mathcal {O}}}^1\left( L_{w_0}, \theta ^\mu _0\Delta (y\cdot \mu )\langle i \rangle \right) =0. \end{aligned}$$

For this, we consider the short exact sequence

$$\begin{aligned} 0\rightarrow \theta ^\mu _0\Delta (y\cdot \mu )\langle \ell (w_0y) \rangle \rightarrow I_{w_0}\rightarrow M\rightarrow 0 \end{aligned}$$

which induces

$$\begin{aligned}&\hom \left( L_{w_0},I_{w_0}\langle -\ell (w_0y)+i\rangle \right) \rightarrow \hom \left( L_{w_0}, M\langle -\ell (w_0y)+i\rangle \right) \\&\quad \rightarrow {\text {ext}}_{{\mathcal {O}}}^1\left( L_{w_0}, \theta ^\mu _0\Delta (y\cdot \mu )\langle i \rangle \right) \rightarrow 0. \end{aligned}$$

Thus, it remains to show that $\hom (L_{w_0}, M\langle j \rangle )=0$, for $j=-\ell (w_0y)+i \ne -2$. We denote by K the cokernel of the inclusion $\theta ^\mu _0\Delta (y\cdot \mu )\langle \ell (w_0y)\rangle \subset \theta ^\mu _0\Delta (\mu )\langle \ell (w_0)\rangle$. In particular, the multiplicity of any shift of $L_{w_0}$ in K is zero. Then we have a canonical map $K\hookrightarrow M$ by definition of K and M. This defines a short exact sequence

$$\begin{aligned} 0 \rightarrow K \rightarrow M\rightarrow Q\rightarrow 0, \end{aligned}$$

where Q has a filtration by $\Delta _{w}\langle \ell (w_0)+\ell (w) \rangle$, for each $w\in S_n{\setminus } S_I$.

We claim that the socle of Q comes from the socle of $\Delta _s\langle \ell (w_0)+1 \rangle$, for $s\in S{\setminus } I$. Indeed, The module $I_{w_0}$ is Koszul dual to the projective resolution of $L_{e}$. The latter can be obtained from the BGG resolution of $L_{e}$, given in terms of Verma modules, by gluing projective resolutions of Verma constituents of the BGG resolution into a projective resolution of $L_e$. In this way, the Verma modules in the BGG resolution give rise to the subquotients of the dual Verma filtration of $I_{w_0}$. Note that any $\Delta _x$ appears in the BGG resolution once and, moreover, for any y such that $\ell (y)=\ell (x)+1$ and $y>x$, the map $\Delta _y\langle 1\rangle \rightarrow \Delta _x$ in the BGG resolution is non-zero, see [6]. The Koszul dual of this property is that the corresponding subquotient of the dual Verma flag of $I_{w_0}$ is given by a nonzero morphism in the derived category and thus by a non-split short exact sequence

$$\begin{aligned} 0\rightarrow \nabla _{w_0x^{-1}}\langle -1\rangle \rightarrow R\rightarrow \nabla _{w_0y^{-1}}\rightarrow 0. \end{aligned}$$

Applying the simple preserving duality $\star$ on $\mathcal {O}$, we consider the dual short exact sequence

$$\begin{aligned} 0\rightarrow \Delta _{w_0y^{-1}}\rightarrow R^\star \rightarrow \Delta _{w_0x^{-1}}\langle 1\rangle \rightarrow 0. \end{aligned}$$

(17)

Write $w_0x^{-1}=u_1tu_2$ reduced, where t is a simple reflection and $w_0y^{-1}=u_1u_2$. Then we can consider (17) as the image, under the $u_2$-shuffling, of a non-split short exact sequence

$$\begin{aligned} 0\rightarrow \Delta _{u_1}\rightarrow R_1\rightarrow \Delta _{u_1t}\langle 1\rangle \rightarrow 0. \end{aligned}$$

(18)

If $w_0=u_3u_1t$ is reduced, then, twisting (18) by $u_3$, we obtain a non-split short exact sequence

$$\begin{aligned} 0\rightarrow \Delta _{w_0t}\rightarrow R_2\rightarrow \Delta _{w_0}\langle 1\rangle \rightarrow 0. \end{aligned}$$

(19)

Here $\Delta _{w_0}=L_{w_0}$ and hence the fact that (19) is non-split means that $R_2$ has simple socle. In particular, the action of the center of $\mathcal {O}_0$ on $R_2$ is not semi-simple. This implies that the action of the center of $\mathcal {O}_0$ on both $R_1$ and $R^\star$ is not semi-simple either and hence they both have simple socle. Since $I_{w_0}$ is self-dual with respect to $\star$, this implies that the socle of Q comes from its Bruhat minimal subquotients $\Delta _s\langle \ell (w_0)+1 \rangle$, where $s\in S{\setminus } I$.

Applying $\hom (L_{w_0},-)$, we obtain

$$\begin{aligned} \hom \left( L_{w_0},M \langle j \rangle \right) \hookrightarrow \hom \left( L_{w_0}, Q\langle j \rangle \right) . \end{aligned}$$

However,

$$\begin{aligned} \hom \left( L_{w_0}, Q\langle j \rangle \right) =\hom \left( L_{w_0}, \bigoplus _{s\in S{\setminus } I}\Delta _s \langle \ell (w_0)+1 +j \rangle \right) =0, \end{aligned}$$

for $j\ne -2$. This completes the proof. $\square$

4 Representation theory versus combinatorics

4.1 Tetrahedron

Our computation for KL polynomials for $\mathcal {J}$ in Proposition 12 gives a nice geometric diagram for simple subquotients of $\Delta _e$ of the form $L_w\langle - i\rangle$, where $w\in \mathcal {J}$ and $i\in \mathbb {Z}$. By representing each simple subquotient $L_{w_{i,j}}\langle - k\rangle$ of $\Delta _e$ as the point (i, j, k), we get a tetrahedron in $\mathbb {Z}^3$. In this picture, we join two points if the corresponding subquotients extend in $\Delta _e$ (the existence of the extension follows from the arguments in Sect. 2.4). In Fig. 2 we show explicit examples for $n=3,4,5$ (note that, as usual in depicting positively graded algebras, the z-axis is reversed in the pictures, so that the bottom of the picture consists of all elements of the form $sw_0$, for s a simple reflection).

Fig. 2

The tetrahedron for $n= 3,4,5$, respectively

Via the bijection given by Theorem 1(ii), this gives the Hasse diagram for $({\mathbf {B}}_n,\le )$, cf. [12, Figure 1 and 2].

In Figs. 3 and 4, we give some examples of composition factors of the form $L_x$, where $x\in \mathcal {J}$, in the modules $\Delta _e/\Delta _w$, for $n=4$ and 5. The composition factors of this form in $\Delta _e/\Delta _w$ which are not in the socle are given by the black points in the figures and the composition factors in the socle are marked by their coordinates in the figures. The white points correspond to the composition factors in $\Delta _w$ (and hence outside of $\Delta _e/\Delta _w$).

Fig. 3

Socles of the quotients $\Delta _e/\Delta _w$ for $n=4$, for $w=s_1s_2s_1, s_1s_2s_3, s_2s_3s_1$, respectively, in the first row, and $w=s_1s_2s_3s_2, s_1s_2s_3s_2s_1, s_1s_2s_3s_1s_2$, respectively, in the second row

Fig. 4

Socles of the quotients $\Delta _e/\Delta _w$ for $n=5$, for $w=s_3s_4s_1s_2s_3s_2s_1$ and $w=s_1s_2s_3s_4s_2s_3s_1s_2$, respectively

For convenience of the reader, on Fig. 5 we present the penultimate cell for $n=5$. In the figure, an element on the position (i, j) has i as the unique left ascent, and j as the unique right ascent.

Fig. 5

The penultimate cell for $n=5$

4.2 Explicit formulae

The set ${\mathbf {B}}_n$ can be explicitly parameterized by triples of integers

$$\begin{aligned} (i,j,k)\quad \text { such that } \quad 1\le & {} i, j \le n-1\quad \text { and }\quad 0 \nonumber \\\le & {} k \le \min \{i-1,j-1,n-1-i,n-1-j\}. \end{aligned}$$

For $i \le j$, put:

$$\begin{aligned} b(i,j,k) := \left( s_is_{i+1} \ldots s_{j+k}\right) \left( s_{i-1}s_i \ldots s_{j+k-1}\right) \cdots \left( s_{i-k}s_{i-k+1} \ldots s_j\right) , \end{aligned}$$

and if $j<i$, set $b(i,j,k) := b(j,i,k)^{-1}$.

We can draw $w\in S_n$ as a picture on a two-dimensional grid, in the following way: View $w\in S_n$ as a permutation on $\{1,\ldots , n\}$ acting on the left, and put on positions (i, w(i)), $i=1,\ldots , n$, in matrix coordinates. This way, we can visualize bigrassmannian permutations as on Fig. 6.

Fig. 6

The bigrassmannian b(i, j, k). Here $r= \min \{i-1,j-1\}-k$

Proposition 17

The bijection from Theorem 1(ii) is given by

$$\begin{aligned} b(i,j,k) \mapsto L_{w_{i,j}} \left\langle -\frac{(n-1)(n-2)}{2}- |i-j|-2k \right\rangle \end{aligned}$$

Proof

The assignment in the claim does define a bijection by Proposition 12, so we need to check that it agrees with the bijection in Theorem 1. From the definition, we see that $b(i,j,k)<b(i,j,k')$ (in the Bruhat order), for $k<k'$. This property, since the relevant subquotients appear multiplicity-freely, determines the bijection. Since the inclusion of the Verma modules $\Delta _x$ agree with (the opposite of) the Bruhat order, the bijection in Theorem 1 also has this property and thus agrees with the one given in our claim. $\square$

4.3 Essential set and rank of a permutation

For $w \in S_n$, the so-called essential set attached to w (defined in [14]) is given by

$$\begin{aligned} {\text {Ess}}(w) := \left\{ (i,j) :1 \le i,j \le n-1, \ i<w^{-1}(j), \ j<w(i), \ w(i+1) \le j, \ w^{-1}(j+1) \le i \right\} . \end{aligned}$$

In Fig. 7 we give examples of essential sets for $w \in S_5$. To easily find ${\text {Ess}}(w)$, following [22], we put on positions (i, w(i)), $i=1,\ldots , n$, in matrix coordinates, and kill all cells to the right or below of these. The surviving cells are denoted by , and are sometimes called the diagram of w. The south-east corners of the diagram (denoted by ) constitute the essential set attached to w.

Fig. 7

Essential sets attached to $s_3s_4s_1s_2s_3s_2s_1$ and $s_1s_2s_3s_4s_2s_3s_1s_2$ (for $n=5$), respectively

In [22] it is shown that there is a bijection between $\mathbf {BM}(w)$ and ${\text {Ess}}(w)$. More precisely, an element $(i,j) \in {\text {Ess}}(w)$ corresponds to a certain monotone triangle (see the cited article for definition), denoted by $J_{i,k,j+1}$, for some k, which is then identified with a bigrassmannian permutation. From the description of monotone triangles in [30, Section 8], it follows that $J_{i,k,j+1}$ correspond to a bigrassmannian permutation with left descent j and right descent i. From Proposition 14 we know that such a bigrassmannian element in $\mathbf {BM}(w)$ is uniquely determined. We conclude:

Corollary 18

(A reformulation of [22, Theorem]) For $w \in S_n$, the map

$$\begin{aligned} x \mapsto (\text {right descent of }x, \text {left descent of }x) \end{aligned}$$

is a bijection $\mathbf {BM}(w) \rightarrow {\text {Ess}}(w)$.

The following result allows us to determine the socle of $\Delta _e/\Delta _w$ via the essential set of w which is very easy and efficient to compute.

Corollary 19

The (ungraded) socle of $\Delta _e/\Delta _w$ is given by $\displaystyle \bigoplus \nolimits _{(i,j) \in {\text {Ess}}(w)} L_{w_{j,i}}$.

Proof

The claim follows from Proposition 17 and Corollary 18. $\square$

To illustrate how this corollary works, one can compare Figs. 7 with Fig. 4 using Fig. 5. Observe that the essential set alone does not provide information on the degrees of the composition factors in the socle of $\Delta _e/\Delta _w$. To get these degrees, we need another combinatorial tool called the rank function.

The rest of the subsection provides an upgrade of the description in Corollary 19 to the graded setup. For $w \in S_n$, the so-called rank function $r_w$ (defined in [14]) is given by:

$$\begin{aligned} r_w(i,j) := |\{k \le i :w(k) \le j \}|, \qquad 1 \le i,j \le n. \end{aligned}$$

More useful for us is the function $t_w$, which we call the co-rank function, given by

$$\begin{aligned} t_w(i,j) := \min \{i,j\}-r_w(i,j). \end{aligned}$$

If we again consider a permutation as a picture on a two-dimensional grid as before, then $r_w(i,j)$ is equal to the number of in the north-west area in Fig. 8. If $i \le j$, then $t_w(i,j)$ is equal to the number of in the north-east area in Fig. 8, and otherwise to the number of in the south-west area.

Fig. 8

Areas which determine $r_w(i,j)$ and $t_w(i,j)$

The co-rank functions for the examples in Fig. 7 are given in Fig. 9.

Fig. 9

The values of $t_w(i,j)$ for $w=s_3s_4s_1s_2s_3s_2s_1$ and $w=s_1s_2s_3s_4s_2s_3s_1s_2$ (for $n=5$), respectively. The values at ${\text {Ess}}(w)$ are boxed

Lemma 20

For $w,x\in S_n$, we have $t_w\le t_x$ if and only if $w\le x$.

Proof

See [4, Theorem 2.1.5]. $\square$

The following lemma is clear from Fig. 6:

Lemma 21

For each $b(i,j,k)\in {\mathbf {B}}_n$, we have ${\text {Ess}}(b(i,j,k))=\{(j,i)\}$ and $t_{b(i,j,k)}(j,i)=k+1$.

Finally, we can describe the graded shifts of the socle constituents in $\Delta _e/(\Delta _w\langle - \ell (w)\rangle )$.

Proposition 22

Let $w\in S_n$. The socle of $N_w:=\Delta _e/(\Delta _w\langle - \ell (w)\rangle )$ in $^{\mathbb {Z}} {\mathcal {O}}$ is given by

$$\begin{aligned} \bigoplus _{(i,j)\in {\text {Ess}}(w)} L_{w_{j,i}}\left\langle -\frac{(n-1)(n-2)}{2}-|i-j|-2(t_w(i,j)-1) \right\rangle . \end{aligned}$$

Proof

Note first that, for $w=b(j,i,k)\in {\mathbf {B}}_n$, we have

$$\begin{aligned} {\text {soc}}(N_w) =L_{w_{j,i}}\langle -{\mathbf {a}}(\mathcal {J})-|i-j|-2(t_w(i,j)-1) \rangle \end{aligned}$$

(20)

by Lemma 21 and Proposition 17. Recall that ${\mathbf {a}}(\mathcal {J})=\frac{(n-1)(n-2)}{2}$.

Now let $w\in S_n$ be arbitrary and let $(i,j)\in {\text {Ess}}(w)$. By Corollary 19 we only need to determine the degree shift, say m, of $L_{w_{j,i}}$ in the socle of $N_w$. Let b(j, i, k) be the element in $\mathbf {BM}(w)$ which corresponds to (i, j) under the bijection in Corollary 18. Since $b(j,i,k)\in \mathbf {BM}(w)$, by Lemma 20 we have $b(j,i,k)\le w$ while $b(j,i,k+1)\not \le w$, and thus $N_w\not \rightarrow \mathrel {}\rightarrow N_{b(j,i,k+1)}$ while $N_w \rightarrow \mathrel {}\rightarrow N_{b(j,i,k)}$. So $m = {\mathbf {a}}(\mathcal {J})+|i-j|+2(t_w(i,j)-1)$ follows from (20) and the parity of m. $\square$

5 Further remarks

5.1 Inclusions between arbitrary Verma modules

An immediate consequence of Theorem 1 is:

Corollary 23

Let $v,w\in S_n$ be such that $v<w$.

• (i) The bijection from Theorem 1(ii) induces a bijection between simple subquotients of $\Delta _v/\Delta _w$ of the form $L_x$, where $x\in \mathcal {J}$, and $y\in {\mathbf {B}}_n$ such that $y\le w$ and $y\not \le v$.

• (ii) The socle of $\Delta _v/\Delta _w$ consists of all $L_x$, where x corresponds to an element in $\mathbf {BM}(w){\setminus } \mathbf {BM}(v)$.

A more detailed description of the socle of $\Delta _v\langle - \ell (v)\rangle /(\Delta _w\langle - \ell (w)\rangle )$ as an object in $^{\mathbb {Z}}{\mathcal {O}}_0$ follows from Proposition 22.

5.2 No such clean result in other types

Unfortunately, Theorem 1 is not true, in general, in other types. One of the reasons is that Corollary 13 fails already in types $B_3$ and $D_4$. We note that ${\mathbf {B}}_n$ agrees in type A with the set of join-irreducible elements in W, while, in general, there are bigrassmannian elements that are not join-irreducible. To generalize Theorem 1, we need, to start with, replace ${\mathbf {B}}_n$ by the set of join-irreducible elements in W. But even then, most of our crucial arguments fail outside type A. For example, in non-simply laced types, for some pairs of simple reflections s and t there will be more than one element $w\in \mathcal {J}$ such that $sw>w$ and $wt>w$. Another problem is that neither bigrassmannian elements nor join-irreducible elements with fixed left and right descents form a chain with respect to the Bruhat order.

Rank two case is, however, special. In this case $\mathcal {J}$ is the set of bigrassmannian elements and all KL polynomials are trivial. So, Theorem 1 is true. Notably, an appropriate analogue of the map $\Phi$ in this case is not the identity map.