Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

2 Preliminaries

In this section, n denotes a nonnegative integer. Define $[n]$ to be $\{1,2,\ldots , n\}$ if $n> 0$ or $\emptyset $ otherwise. In addition, we set $[-1]:=\emptyset $ . For positive integers $i\le j$ , set $[i,j]:=\{i,i+1,\ldots , j\}$ .

2.1 Compositions and their diagrams

A composition $\alpha $ of a nonnegative integer n, denoted by $\alpha \models n$ , is a finite ordered list of positive integers $(\alpha _1, \alpha _2, \ldots , \alpha _k)$ satisfying $\sum _{i=1}^k \alpha _i = n$ . For each $1 \le i \le k$ , let us call $\alpha _i$ a part of $\alpha $ . And we call $k =: \ell (\alpha )$ the length of $\alpha $ and $n =:|\alpha |$ the size of $\alpha $ . For convenience, we define the empty composition $\emptyset $ to be the unique composition of size and length $0$ . A generalised composition ${\boldsymbol {\unicode{x3b1} }}$ of n is a formal sum $\alpha ^{(1)} \oplus \alpha ^{(2)} \oplus \cdots \oplus \alpha ^{(k)}$ , where $\alpha ^{(i)} \models n_i$ for positive integers $n_i$ s with $n_1 + n_2 + \cdots + n_k = n$ .

For $\alpha = (\alpha _1, \alpha _2, \ldots , \alpha _{\ell (\alpha )}) \models n$ , we define the composition diagram $\mathtt {cd}(\alpha )$ of $\alpha $ as a left-justified array of n boxes where the ith row from the top has $\alpha _i$ boxes for $1 \le i \le k$ . We also define the ribbon diagram $\mathtt {rd}(\alpha )$ of $\alpha $ by the connected skew diagram without $2 \times 2$ boxes, such that the ith column from the left has $\alpha _i$ boxes. Then for a generalised composition ${\boldsymbol {\unicode{x3b1} }}$ of n, we define the generalised ribbon diagram $\mathtt {rd}({\boldsymbol {\unicode{x3b1} }})$ of ${\boldsymbol {\unicode{x3b1} }}$ to be the skew diagram whose connected components are $\mathtt {rd}(\alpha ^{(1)}), \mathtt {rd}(\alpha ^{(2)}), \ldots , \mathtt {rd}(\alpha ^{(k)})$ such that $\mathtt {rd}(\alpha ^{(i+1)})$ is strictly to the northeast of $\mathtt {rd}(\alpha ^{(i)})$ for $i = 1, 2, \ldots , k-1$ . For example, if $\alpha = (3,1,2)$ and ${\boldsymbol {\unicode{x3b1} }} = (2,1) \oplus (1,1)$ , then

Given $\alpha = (\alpha _1, \alpha _2, \ldots ,\alpha _{\ell (\alpha )}) \models n$ and $I = \{i_1 < i_2 < \cdots < i_k\} \subset [n-1]$ , let

$$ \begin{align*} &\mathrm{set}(\alpha) := \{\alpha_1,\alpha_1+\alpha_2,\ldots, \alpha_1 + \alpha_2 + \cdots + \alpha_{\ell(\alpha)-1}\}, \\ &\mathrm{comp}(I) := (i_1,i_2 - i_1,\ldots,n-i_k). \end{align*} $$

The set of compositions of n is in bijection with the set of subsets of $[n-1]$ under the correspondence $\alpha \mapsto \mathrm {set}(\alpha )$ (or $I \mapsto \mathrm {comp}(I)$ ). Let $\alpha ^{\mathrm {r}}$ denote the composition $(\alpha _{\ell (\alpha )}, \alpha _{\ell (\alpha )-1}, \ldots , \alpha _1)$ .

For compositions $\alpha = (\alpha _{1}, \alpha _{2}, \ldots , \alpha _{k})$ and $\beta = (\beta _{1}, \beta _{2}, \ldots , \beta _{l})$ , let $\alpha \cdot \beta $ be the concatenation and $\alpha \odot \beta $ the near concatenation of $\alpha $ and $\beta $ . In other words, $ \alpha \cdot \beta = (\alpha _1, \alpha _2, \ldots , \alpha _k, \beta _1, \beta _2, \ldots , \beta _l)$ and $\alpha \odot \beta = (\alpha _1,\ldots , \alpha _{k-1},\alpha _k + \beta _1,\beta _2, \ldots , \beta _l)$ . For a generalised composition ${\boldsymbol {\unicode{x3b1} }} = \alpha ^{(1)} \oplus \alpha ^{(2)} \oplus \cdots \oplus \alpha ^{(m)}$ , define

$$ \begin{align*} [{\boldsymbol{\unicode{x3b1}}}] := \{\alpha^{(1)} \ \square \ \alpha^{(2)} \ \square \ \cdots \ \square \ \alpha^{(m)} \mid \square = \cdot \text{ or } \odot\}. \end{align*} $$

2.2 The $0$ -Hecke algebra and the quasisymmetric characteristic

The symmetric group $\Sigma _n$ is generated by simple transpositions $s_i := (i \ i \hspace {-.5ex} + \hspace {-.5ex} 1)$ with $1 \le i \le n-1$ . An expression for $\sigma \in \Sigma _n$ of the form $s_{i_1} s_{i_2} \cdots s_{i_p}$ that uses the minimal number of simple transpositions is called a reduced expression for $\sigma $ . The number of simple transpositions in any reduced expression for $\sigma $ , denoted by $\ell (\sigma )$ , is called the length of $\sigma $ .

The $0$ -Hecke algebra $H_n(0)$ is the $\mathbb C$ -algebra generated by $\pi _1, \pi _2, \ldots ,\pi _{n-1}$ subject to the following relations:

$$ \begin{align*} \pi_i^2 &= \pi_i \quad \text{for } 1\le i \le n-1,\\ \pi_i \pi_{i+1} \pi_i &= \pi_{i+1} \pi_i \pi_{i+1} \quad \text{for } 1\le i \le n-2,\\ \pi_i \pi_j &=\pi_j \pi_i \quad \text{if } |i-j| \ge 2. \end{align*} $$

Pick up any reduced expression $s_{i_1} s_{i_2} \cdots s_{i_p}$ for a permutation $\sigma \in \Sigma _n$ . It is well known that the element $\pi _{\sigma } := \pi _{i_1} \pi _{i_2} \cdots \pi _{i_p}$ is independent of the choice of reduced expressions and $\{\pi _{\sigma } \mid \sigma \in \Sigma _n\}$ is a basis for $H_n(0)$ . For later use, set

$$ \begin{align*} \pi_{[i, j]} := \pi_{i} \pi_{i+1} \cdots \pi_{j} \quad \text{and} \quad \pi_{[i, j]^{\mathrm{r}}} := \pi_{j} \pi_{j-1} \cdots \pi_{i} \end{align*} $$

for all $1 \le i \le j \le n-1$ .

Let $\mathcal {R}(H_n(0))$ denote the $\mathbb Z$ -span of (representatives of) the isomorphism classes of finite-dimensional representations of $H_n(0)$ . The isomorphism class corresponding to an $H_n(0)$ -module M will be denoted by $[M]$ . The Grothendieck group $\mathcal {G}_0(H_n(0))$ is the quotient of $\mathcal {R}(H_n(0))$ modulo the relations $[M] = [M'] + [M"]$ whenever there exists a short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M" \rightarrow 0$ . The equivalence classes of irreducible representations of $H_n(0)$ form a free $\mathbb Z$ -basis for $\mathcal {G}_0(H_n(0))$ . Let

$$\begin{align*}\mathcal{G} := \bigoplus_{n \ge 0} \mathcal{G}_0(H_n(0)). \end{align*}$$

According to [Reference Norton25], there are $2^{n-1}$ distinct irreducible representations of $H_n(0)$ . They are naturally indexed by compositions of n. Let $\mathbf {F}_{\alpha }$ denote the $1$ -dimensional $\mathbb C$ -vector space corresponding to $\alpha \models n$ , spanned by a vector $v_{\alpha }$ . For each $1\le i \le n-1$ , define an action of the generator $\pi _i$ of $H_n(0)$ as follows:

$$\begin{align*}\pi_i \cdot v_{\alpha} = \begin{cases} 0 & i \in \mathrm{set}(\alpha),\\ v_{\alpha} & i \notin \mathrm{set}(\alpha). \end{cases} \end{align*}$$

Then $\mathbf {F}_{\alpha }$ is an irreducible $1$ -dimensional $H_n(0)$ -representation.

In the following, let us review the connection between $\mathcal {G}$ and the ring $\mathrm {QSym}$ of quasisymmetric functions. Quasisymmetric functions are power series of bounded degree in variables $x_{1},x_{2},x_{3},\ldots $ with coefficients in $\mathbb Z$ that are shift invariant in the sense that the coefficient of the monomial $x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{k}^{\alpha _{k}}$ is equal to the coefficient of the monomial $x_{i_{1}}^{\alpha _{1}}x_{i_{2}}^{\alpha _{2}}\cdots x_{i_{k}}^{\alpha _{k}}$ for any strictly increasing sequence of positive integers $i_{1}<i_{2}<\cdots <i_{k}$ indexing the variables and any positive integer sequence $(\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})$ of exponents.

Given a composition $\alpha $ , the fundamental quasisymmetric function $F_{\alpha }$ is defined by $F_{\emptyset } = 1$ and

$$\begin{align*}F_{\alpha} = \sum_{\substack{1 \le i_1 \le i_2 \le \cdots \le i_k \\ i_j < i_{j+1} \text{ if } j \in \mathrm{set}(\alpha)}} x_{i_1} x_{i_2} \cdots x_{i_k}. \end{align*}$$

It is well known that $\{F_{\alpha } \mid \alpha \text { is a composition}\}$ is a basis for $\mathrm {QSym}$ . In [Reference Duchamp, Krob, Leclerc and Thibon15], Duchamp, Krob, Leclerc and Thibon show that, when $\mathcal {G}$ is equipped with induction product, the linear map

$$ \begin{align*} \mathrm{ch} : \mathcal{G} \rightarrow \mathrm{QSym}, \quad [\mathbf{F}_{\alpha}] \mapsto F_{\alpha}, \end{align*} $$

called the quasisymmetric characteristic, is a ring isomorphism.

2.3 Projective modules of the $0$ -Hecke algebra

We begin this subsection by recalling that $H_n(0)$ is a Frobenius algebra. Hence it is self-injective, so that finitely generated projective and injective modules coincide (see [Reference Duchamp, Hivert and Thibon14, Proposition 4.1], [Reference Fayers16, Proposition 4.1] and [Reference Benson3, Proposition 1.6.2]).

It was Norton [Reference Norton25] who first classified all projective indecomposable modules of $H_n(0)$ up to isomorphism that bijectively correspond to compositions of n. Later, Huang [Reference Huang20] provided a combinatorial description of these modules and their induction products as well by using standard ribbon tableaux of generalised composition shape. We review Huang’s description very briefly here.

Definition 2.1. For a generalised composition ${\boldsymbol {\unicode{x3b1} }}$ of n, a standard ribbon tableau (SRT) of shape ${\boldsymbol {\unicode{x3b1} }}$ is a filling of $\mathtt {rd}({\boldsymbol {\unicode{x3b1} }})$ with $\{1,2,\ldots ,n\}$ such that the entries are all distinct, the entries in each row are increasing from left to right, and the entries in each column are increasing from top to bottom.

Let $\mathrm {SRT}({\boldsymbol {\unicode{x3b1} }})$ denote the set of all $\mathrm {SRT}$ x of shape ${\boldsymbol {\unicode{x3b1} }}$ . For $T \in \mathrm {SRT}({\boldsymbol {\unicode{x3b1} }})$ , let

$$ \begin{align*} \mathrm{Des}(T) := \{i \in [n-1] \mid i \text{ appears weakly below } i+1 \text{ in } T \}. \end{align*} $$

Define an $H_n(0)$ -action on the $\mathbb C$ -span of $\mathrm {SRT}({\boldsymbol {\unicode{x3b1} }})$ by

(2.1) $$ \begin{align} \pi_i \cdot T = \begin{cases} T & \text{if } i \notin \mathrm{Des}(T),\\ 0 & \text{if } i \text{ and } i+1 \text{ are in the same row of } T,\\ s_i \cdot T & \text{if } i \text{ appears strictly below } i+1 \text{ in } T \end{cases} \end{align} $$

for $1\le i \le n-1$ and $T \in \mathrm {SRT}({\boldsymbol {\unicode{x3b1} }})$ . Here $s_i \cdot T$ is obtained from T by swapping i and $i+1$ . The resulting module is denoted by $\mathbf {P}_{\boldsymbol {\unicode{x3b1} }}$ . It is known that the set $\{\mathbf {P}_{\alpha } \mid \alpha \models n\}$ forms a complete family of non-isomorphic projective indecomposable $H_n(0)$ -modules and $\mathbf {P}_{\alpha } /\mathrm {rad}(\mathbf {P}_{\alpha }) \cong \mathbf {F}_{\alpha }$ , where $\mathrm {rad}(\mathbf {P}_{\alpha })$ is the radical of $\mathbf {P}_{\alpha }$ (for details; see [Reference Huang20, Reference Norton25]).

Remark 2.2. It should be pointed out that the ribbon diagram and $H_n(0)$ -action used here are slightly different from those in Huang’s work [Reference Huang20]. He describes the $H_n(0)$ -action on $\mathbf {P}_{\boldsymbol {\unicode{x3b1} }}$ in terms of $\overline {\pi }_i$ s, where $\overline {\pi }_i= \pi _i -1$ . On the other hand, we use $\pi _i$ s because the $H_n(0)$ -action on $\mathcal {V}_{\alpha }$ is described in terms of $\pi _i$ s. This leads us to adjust Huang’s ribbon diagram to the form of $\mathtt {rd}({\boldsymbol {\unicode{x3b1} }})$ .

Given any generalised composition ${\boldsymbol {\unicode{x3b1} }}$ , let $T_{\boldsymbol {\unicode{x3b1} }} \in \mathrm {SRT}({\boldsymbol {\unicode{x3b1} }})$ be the $\mathrm {SRT}$ obtained by filling $\mathtt {rd}({\boldsymbol {\unicode{x3b1} }})$ with entries $1, 2, \ldots , n$ from top to bottom and from left to right. Since $\mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}}$ is cyclically generated by $T_{\boldsymbol {\unicode{x3b1} }}$ , we call $T_{\boldsymbol {\unicode{x3b1} }}$ the source tableau of $\mathbf {P}_{\boldsymbol {\unicode{x3b1} }}$ . For any $\mathrm {SRT} T$ , let $\mathbf {w}(T)$ be the word obtained by reading the entries from left to right, starting with the bottom row. Using this reading, Huang [Reference Huang20] shows the following result.

Theorem 2.3 ([Reference Huang20, Theorem 3.3]).

Let ${\boldsymbol {\unicode{x3b1} }}$ be a generalised composition of n. Then $\mathbf {P}_{\boldsymbol {\unicode{x3b1} }}$ is isomorphic to $\bigoplus _{\beta \in [{\boldsymbol {\unicode{x3b1} }}]} \mathbf {P}_{\beta }$ as an $H_n(0)$ -module.

For later use, for every generalised composition ${\boldsymbol {\unicode{x3b1} }}$ of n, we define a partial order $\le $ on $\mathrm {SRT}({\boldsymbol {\unicode{x3b1} }})$ by

$$ \begin{align*} T \le T' \quad \text{if and only if} \quad T' = \pi_{\sigma} \cdot T \quad \text{for some } \sigma \in \Sigma_n. \end{align*} $$

As usual, whenever $T \le T'$ , the notation $[T, T']$ denotes the interval $\{U \in \mathrm {SRT}({\boldsymbol {\unicode{x3b1} }}) \mid T \le U \le T'\}$ .

2.4 The $H_n(0)$ -action on standard immaculate tableaux

Noncommutative Bernstein operators were introduced by Berg, Bergeron, Saliola, Serrano and Zabrocki [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki4]. Applied to the identity of the ring $\mathrm {NSym}$ of noncommutative symmetric functions, they yield the immaculate functions, which form a basis of $\mathrm {NSym}$ . Soon after, using the combinatorial objects called standard immaculate tableaux, they constructed indecomposable $H_n(0)$ -modules whose quasisymmetric characteristics are the quasisymmetric functions that are dual to immaculate functions (see [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki5]).

Definition 2.4. Let $\alpha \models n$ . A standard immaculate tableau (SIT) of shape $\alpha $ is a filling $\mathscr {T}$ of the composition diagram $\mathtt {cd}(\alpha )$ with $\{1,2,\ldots ,n\}$ such that the entries are all distinct, the entries in each row increase from left to right, and the entries in the first column increase from top to bottom.

We denote the set of all SITx of shape $\alpha $ by $\mathrm {SIT}(\alpha )$ . For $\mathscr {T} \in \mathrm {SIT}(\alpha )$ , let

$$ \begin{align*} \mathrm{Des}(\mathscr{T}) := \{i \in [n-1] \mid i \text{ appears strictly above } i+1 \text{ in } \mathscr{T} \}. \end{align*} $$

Define an $H_n(0)$ -action on $\mathbb C$ -span of $\mathrm {SIT}(\alpha )$ by

(2.2) $$ \begin{align} \pi_i \cdot \mathscr{T} = \begin{cases} \mathscr{T} & \text{if } i \notin \mathrm{Des}(\mathscr{T}),\\ 0 & \text{if } i \text{ and } i+1 \text{ are in the first column of } \mathscr{T},\\ s_i \cdot \mathscr{T} & \text{otherwise} \end{cases} \end{align} $$

for $1\le i \le n-1$ and $\mathscr {T} \in \mathrm {SIT}(\alpha )$ . Here $s_i \cdot \mathscr {T}$ is obtained from $\mathscr {T}$ by swapping i and $i+1$ . The resulting module is denoted by $\mathcal {V}_{\alpha }$ .

Let $\mathscr {T}_{\alpha } \in \mathrm {SIT}(\alpha )$ be the SIT obtained by filling $\mathtt {cd}(\alpha )$ with entries $1, 2, \ldots , n$ from left to right and from top to bottom.

Theorem 2.5 ([Reference Berg, Bergeron, Saliola, Serrano and Zabrocki5]).

For $\alpha \models n$ , $\mathcal {V}_{\alpha }$ is a cyclic indecomposable $H_n(0)$ -module generated by $\mathscr {T}_{\alpha }$ whose quasisymmetric characteristic is the dual immaculate quasisymmetric function $\mathfrak {S}^*_{\alpha }$ .

Convention. Regardless of a ribbon diagram or composition diagram, columns are numbered from left to right. To avoid possible confusion, we adopt the following notation:

1. (i) Let T be a filling of the ribbon diagram $\mathtt {rd}({\boldsymbol {\unicode{x3b1} }})$ .

   • - $T^i_j$ = the entry at the ith box from the top of the jth column

   • - $T_j^{-1}$ = the entry at the bottom-most box in the jth column

   • - $T^{\bullet }_j$ = the set of all entries in the jth column

2. (ii) Let $\mathscr {T}$ be a filling of the composition diagram $\mathtt {cd}(\alpha )$ .

   • - $\mathscr {T}_{i,j}$ = the entry at the box in the ith row (from the top) and in the jth column

3 A minimal projective presentation of $\mathcal {V}_{\alpha }$ and $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$

From now on, $\alpha $ denotes an arbitrarily chosen composition of n. We here construct a minimal projective presentation of $\mathcal {V}_{\alpha }$ . Using this, we compute $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$ for each $\beta \models n$ .

Firstly, let us introduce necessary terminologies and notation. Let $A,B$ be finitely generated $H_n(0)$ -modules. A surjective $H_n(0)$ -module homomorphism $f:A\to B$ is called an essential epimorphism if an $H_n(0)$ -module homomorphism $g: X\to A$ is surjective whenever $f \circ g:X\to B$ is surjective. A projective cover of A is an essential epimorphism $f:P\to A$ with P projective that always exists and is unique up to isomorphism. It is well known that $f:P\to A$ is an essential epimorphism if and only if $\ker (f) \subset \mathrm {rad}(P)$ (for instance, see [Reference Auslander, Reiten and Smalø1, Proposition I.3.6]). For simplicity, when f is clear in the context, we just write $\Omega (A)$ for $\ker (f)$ and call it the syzygy module of A. An exact sequence

with projective modules $P_0$ and $P_1$ is called a minimal projective presentation if the $H_n(0)$ -module homomorphisms $\epsilon : P_0 \rightarrow A$ and $\partial _1: P_1 \rightarrow \Omega (A)$ are projective covers of A and $\Omega (A)$ , respectively.

Next, let us review the projective cover of $\mathcal {V}_{\alpha }$ obtained in [Reference Choi, Kim, Nam and Oh12]. Given any $T \in \mathrm {SRT}(\alpha )$ , let $\mathscr {T}_T$ be the filling of $\mathtt {cd}(\alpha )$ given by $(\mathscr {T}_T)_{i,j} = T^{j}_{i}$ . Then we define a $\mathbb C$ -linear map $\Phi : \mathbf {P}_{\alpha } \rightarrow \mathcal {V}_{\alpha }$ by

(3.1) $$ \begin{align} \Phi(T) = \begin{cases} \mathscr{T}_T & \text{if } \mathscr{T}_T \text{ is an SIT,}\\ 0 & \text{otherwise.} \end{cases} \end{align} $$

For example, if $\alpha = (1,2,2)$ and

then

Therefore, $\Phi (T_1) = \mathscr {T}_{T_1}$ and $\Phi (T_2) = 0$ .

Now, let us construct a projective cover of $\Omega (\mathcal {V}_{\alpha })$ for each $\alpha \models n$ . To do this, we provide necessary notation. For each integer $0\le i \le \ell (\alpha )-1$ , we set $m_i$ to be $\sum _{j = 1}^{i}\alpha _j$ for $i> 0$ and $m_0 = 0$ . Let

$$\begin{align*}\mathcal{I}(\alpha) := \{1 \le i \le \ell(\alpha)-1 \mid \alpha_{i+1} \neq 1 \}. \end{align*}$$

Given $i \in \mathcal {I}(\alpha )$ , let

$$\begin{align*}T^{(i)}_{\alpha} := \pi_{[m_{i-1} + 1, m_{i}]} \cdot T_{\alpha} \end{align*}$$

and

$$\begin{align*}{\boldsymbol{\unicode{x3b1}}}^{(i)} := (\alpha_1, \alpha_2, \ldots, \alpha_{i-1}, \alpha_i +1, \alpha_{i+1} - 1) \oplus (\alpha_{i+2} , \alpha_{i+3}, \ldots, \alpha_{\ell(\alpha)}). \end{align*}$$

Given an SRT $\tau $ of shape ${\boldsymbol {\unicode{x3b1} }}^{(i)} (i \in \mathcal {I}(\alpha ))$ , define $L(\tau )$ to be the filling of $\mathtt {rd}(\alpha )$ whose entries in each column are increasing from top to bottom and whose columns are given as follows: for $1 \le p \le \ell (\alpha )$ ,

(3.2) $$ \begin{align} L(\tau)_p^{\bullet} = \begin{cases} \tau_i^{\bullet} \setminus \{\tau_i^1\} & \text{if } p = i,\\ \tau_{i+1}^{\bullet} \cup \{\tau_i^1\} & \text{if } p = i+1,\\ \tau_p^{\bullet} & \text{otherwise.}\\ \end{cases} \end{align} $$

For each $i \in \mathcal {I}(\alpha )$ , we define a $\mathbb C$ -linear map $ \partial _1^{(i)}: \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}^{(i)}} \rightarrow H_n(0) \cdot T^{(i)}_{\alpha }$ by

$$\begin{align*}\partial_1^{(i)} (\tau) = \begin{cases} L(\tau) & \text{if } L(\tau) \in \mathrm{SRT}(\alpha),\\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

Then we define a $\mathbb C$ -linear map $\partial _1 : \bigoplus _{i \in \mathcal {I}(\alpha )} \mathbf {P}_{{\boldsymbol {\unicode{x3b1} }}^{(i)}} \rightarrow \mathbf {P}_{\alpha }$ by

$$ \begin{align*}\partial_1 := \sum_{i \in \mathcal{I}(\alpha)} \partial_1^{(i)}. \end{align*} $$

Example 3.4. Let $\alpha = (1,2,1)$ . Then we have that $\mathcal {I}(\alpha ) = \{1\}$ and ${\boldsymbol {\unicode{x3b1} }}^{(1)} = (2,1) \oplus (1)$ .

(a) The map $\partial _1: \mathbf {P}_{(2,1) \oplus (1)} \rightarrow \mathbf {P}_{(1,2,1)}$ is illustrated in Figure 1, where the entries i in red in each SRT $T$ are used to indicate that $\pi _i \cdot T = 0$ .

Figure 1 $\partial _1: \mathbf {P}_{(2,1) \oplus (1)} \rightarrow \mathbf {P}_{(1,2,1)}$ .

(b) Note that $\mathcal {J}(\alpha ) = [{\boldsymbol {\unicode{x3b1} }}^{(1)}] = \{(2,2),(2,1,1)\}$ . By Theorem 3.3(b), we have

$$\begin{align*}\dim \mathrm{Ext}_{H_n(0)}^1(\mathcal{V}_{(1,2,1)},\mathbf{F}_{\beta}) = \begin{cases} 1 & \text{ if } \beta = (2,2) \text{ or } (2,1,1), \\ 0 & \text{ otherwise.} \end{cases} \end{align*}$$

4 A minimal injective presentation of $\mathcal {V}_{\alpha }$ and $\mathrm {Ext}^1_{H_n(0)}(\mathbf {F}_{\beta },\mathcal {V}_{\alpha })$

As before, $\alpha $ denotes an arbitrarily chosen composition of n. In this section, we construct a minimal injective presentation of $\mathcal {V}_{\alpha }$ . Using this, we compute $\mathrm {Ext}^1_{H_n(0)}(\mathbf {F}_{\beta },\mathcal {V}_{\alpha })$ for each $\beta \models n$ .

Let us introduce necessary terminologies and notation. Let $M,N$ be finitely generated $H_n(0)$ -modules with $N \subsetneq M$ . We say that M is an essential extension of N if $X\cap N \ne 0$ for all nonzero submodules X of M. An injective $H_n(0)$ -module homomorphism $\iota : M \rightarrow \boldsymbol {I}$ with $\boldsymbol {I}$ injective is called an injective hull of M if $\boldsymbol {I}$ is an essential extension of $\iota (M)$ that always exists and is unique up to isomorphism. By [Reference Lam23, Theorem 3.30 and Exercise 3.6.12], it follows that $\boldsymbol {I}$ is an injective hull of M if and only if $\iota (M) \supseteq \mathrm {soc}(\boldsymbol {I})$ . Here $\mathrm {soc}(\boldsymbol {I})$ is the socle of $\boldsymbol {I}$ : that is, the sum of all simple submodules of $\boldsymbol {I}$ . When $\iota $ is clear in the context, we write $\Omega ^{-1}(M)$ for $\mathrm {Coker} (\iota )$ and call it the cosyzygy module of M. An exact sequence

with injective modules $\boldsymbol {I}_0$ and $\boldsymbol {I}_1$ is called a minimal injective presentation if the $H_n(0)$ -module homomorphisms $\iota : M \rightarrow \boldsymbol {I}_0$ and $\partial ^1: \Omega ^{-1}(M) \rightarrow \boldsymbol {I}_1$ are injective hulls of M and $\Omega ^{-1}(M)$ , respectively.

We first describe an injective hull of $\mathcal {V}_{\alpha }$ . Let

$$ \begin{align*} \mathcal{K}(\alpha) := \{1 \leq i \leq \ell(\alpha) \mid \alpha_i> 1\} \cup \{0\}. \end{align*} $$

We write the elements of $\mathcal {K}(\alpha )$ as $k_0:=0 < k_1 < k_2 < \cdots < k_m$ . Let

$$ \begin{align*} \underline{{\boldsymbol{\unicode{x3b1}}}} &:= (\alpha_{k_1} -1) \oplus (\alpha_{k_2}-1) \oplus \cdots \oplus \left((\alpha_{k_m}-1) \odot (1^{\ell(\alpha)})\right) \\ & \ = (\alpha_{k_1} -1) \oplus (\alpha_{k_2}-1) \oplus \cdots \oplus (\alpha_{k_{m-1}}-1) \oplus (\alpha_{k_m},1^{\ell(\alpha)-1}). \end{align*} $$

Let us depict $\mathtt {rd}(\underline {{\boldsymbol {\unicode{x3b1} }}})$ in a pictorial manner. When $j=0$ , we define $\mathtt {S}_{k_0}$ to be the vertical strip consisting of all the boxes in the first column of $\mathtt {cd}(\alpha )$ . For $1 \le j \le m$ , we define $\mathtt {S}_{k_j}$ as the horizontal strip consisting of the boxes in the $k_j$ th row of $\mathtt {cd}(\alpha )$ (from the top), except for the leftmost box. Then $\underline {{\boldsymbol {\unicode{x3b1} }}}$ is defined by the generalised composition obtained by placing $\mathtt {S}_{k_0},\mathtt {S}_{k_1},\ldots , \mathtt {S}_{k_m}$ in the following manner:

1. (i) $\mathtt {S}_{k_0}$ is placed horizontally at the topmost row in the new diagram.

2. (ii) $\mathtt {S}_{k_m}$ is placed vertically to the lower-left of $\mathtt {S}_{k_0}$ so that $\mathtt {S}_{k_0}$ and $\mathtt {S}_{k_m}$ are connected.

3. (iii) For $j=m-1,m-2, \ldots , 1$ , place $\mathtt {S}_{k_j}$ vertically to the lower-left of $\mathtt {S}_{k_{j+1}}$ so that they are not connected to each other.

Figure 2 illustrates the above procedure.

Figure 2 The construction of $\mathtt {rd}( \underline {{\boldsymbol {\unicode{x3b1} }}})$ when $\alpha =(2,1,3^2,1)$ .

For simplicity, we introduce the following notation:

• • For an SIT $\mathscr {T}$ and a subdiagram $\mathtt {S}$ of shape of $\mathscr {T}$ , we denote by $\mathscr {T}(\mathtt {S})$ the set of entries of $\mathscr {T}$ in $\mathtt {S}$ .

• • For an SRT T and a subdiagram $\mathtt {S}$ of shape of T, we denote by $T(\mathtt {S})$ the set of entries of T in $\mathtt {S}$ .

For $\mathscr {T} \in \mathrm {SIT}(\alpha )$ , let ${T^{\mathscr {T}}}$ be the tableau of $\mathtt {rd}(\underline {{\boldsymbol {\unicode{x3b1} }}})$ defined by

$$ \begin{align*} ({T^{\mathscr{T}}})(\mathtt{S}_{k_j}) := \mathscr{T}(\mathtt{S}_{k_j}) \qquad \text{for } 0 \leq j \leq m. \end{align*} $$

Extending the assignment $\mathscr {T} \mapsto T^{\mathscr {T}}$ by linearity, we define the $\mathbb C$ -linear map

$$ \begin{align*} \epsilon: \mathcal{V}_{\alpha} \rightarrow \mathbf{P}_{\underline{{\boldsymbol{\unicode{x3b1}}}}}, \quad \mathscr{T} \mapsto {T^{\mathscr{T}}}, \end{align*} $$

which is obviously injective.

For later use, we provide bases of $\epsilon (\mathcal {V}_{\alpha })$ and $\Omega ^{-1}(\mathcal {V}_{\alpha })$ . From the injectivity of $\epsilon $ , we derive that $\epsilon (\mathcal {V}_{\alpha })$ is spanned by

$$ \begin{align*} \{ T \in \mathrm{SRT}(\underline{{\boldsymbol{\unicode{x3b1}}}}) \mid T_j^{1+\delta_{j,m}}> T^1_{m+k_j-1} \text{ for all } 1\leq j \leq m\} \end{align*} $$

and $\Omega ^{-1}(\mathcal {V}_{\alpha })$ is spanned by $\{T + \epsilon (\mathcal {V}_{\alpha }) \mid T \in \Theta (\mathcal {V}_{\alpha })\}$ with

(4.1) $$ \begin{align} \Theta(\mathcal{V}_{\alpha}) := \{ T \in \mathrm{SRT}(\underline{{\boldsymbol{\unicode{x3b1}}}}) \mid T_j^{1+\delta_{j,m}} < T^1_{m+k_j-1} \text{ for some }1\leq j \leq m \}. \end{align} $$

Example 4.2. If $\alpha = (1,2,2) \models 5$ , then $\mathcal {K}(\alpha ) = \{0,2,3\}$ and $\underline {{\boldsymbol {\unicode{x3b1} }}} = (1)\oplus (2,1^2)$ . For , one sees that . The map $\epsilon :\mathcal {V}_{\alpha } \rightarrow \mathbf {P}_{\underline {{\boldsymbol {\unicode{x3b1} }}}}$ is illustrated in Figure 3, where the red entries i in tableaux are used to indicate that $\pi _i$ acts on them as zero.

Figure 3 $\epsilon : \mathcal {V}_{(1,2,2)} \rightarrow \mathbf {P}_{(1)\oplus (2,1,1)}$ .

We next describe an injective hull of $\Omega ^{-1}(\mathcal {V}_{\alpha })$ . To do this, we need an $H_n(0)$ -module homomorphism $\partial ^1: \mathbf {P}_{\underline {{\boldsymbol {\unicode{x3b1} }}}}\rightarrow \boldsymbol {I}$ with $\boldsymbol {I}$ an injective module satisfying that $\ker (\partial ^1)=\epsilon (\mathcal {V}_{\alpha })$ .

First, we provide the required injective module $\boldsymbol {I}$ . For $1 \leq j \leq m$ , define $\underline {{\boldsymbol {\unicode{x3b1} }}}_{(j)}$ to be the generalised composition

$$ \begin{align*} \underline{{\boldsymbol{\unicode{x3b1}}}}_{(j)}: = \left\{ \begin{array}{ll} (\alpha_{k_1}-1) \oplus \cdots \oplus (\alpha_{k_j} - 2) \oplus \cdots \oplus (\alpha_{k_m}, 1^{\ell(\alpha)-k_j+1}) \oplus (1^{k_j-1}) & \quad \text{if } 1 \leq j < m, \\ (\alpha_{k_1}-1 ) \oplus \cdots \oplus (\alpha_{k_{m-1}}-1) \oplus \left((\alpha_{k_m}-1,1^{\ell(\alpha)-k_j+1}) \cdot (1^{k_j-1})\right) & \quad \text{if } j = m. \end{array} \right. \end{align*} $$

Then we set

(4.2) $$ \begin{align} \boldsymbol{I}:=\bigoplus_{1 \leq j \leq m}\mathbf{P}_{\underline{{\boldsymbol{\unicode{x3b1}}}}_{(j)}}. \end{align} $$

In the following, we provide a pictorial description of $\mathtt {rd}(\underline {{\boldsymbol {\unicode{x3b1} }}}_{(j)})$ . We begin by recalling that $\mathtt {rd}(\underline {{\boldsymbol {\unicode{x3b1} }}})$ consists of the horizontal strip $\mathtt {S}_{k_0}$ and the vertical strips $\mathtt {S}_{k_1},\ldots ,\mathtt {S}_{k_m}$ . For each $-1\le r \le m$ , we denote by $\mathtt {S}^{\prime }_{k_r}$ the connected horizontal strip of length

$$ \begin{align*} |\mathtt{S}^{\prime}_{k_r}|:= \begin{cases} k_j-1 & \text{ if } r = -1,\\ \ell(\alpha)-k_j+2 & \text{ if } r = 0, \\ |\mathtt{S}_{k_r}| - \delta_{r,j} & \text{ if } 1 \leq r \leq m, \end{cases} \end{align*} $$

where $k_{-1} := -1$ . With this preparation, $\underline {{\boldsymbol {\unicode{x3b1} }}}_{(j)}$ is defined to be the generalised composition obtained by placing $\mathtt {S}^{\prime }_{k_{-1}},\mathtt {S}^{\prime }_{k_0},\mathtt {S}^{\prime }_{k_1},\ldots ,\mathtt {S}^{\prime }_{k_m}$ in the following way:

1. (i) $\mathtt {S}^{\prime }_{k_1}$ is placed vertically to the leftmost column in the diagram we will create.

2. (ii) For $j = 2,3,\ldots , m$ , $\mathtt {S}^{\prime }_{k_j}$ is placed vertically to the upper-right of $\mathtt {S}^{\prime }_{k_{j-1}}$ so that they are not connected to each other.

3. (iii) $\mathtt {S}^{\prime }_{k_0}$ is placed horizontally to $\mathtt {S}^{\prime }_{k_m}$ so that they are connected.

4. (iv) In the case where $j \neq m$ , $\mathtt {S}^{\prime }_{k_{-1}}$ is placed horizontally to the upper-right of $\mathtt {S}^{\prime }_{k_0}$ so that they are disconnected. In the case where $j = m$ , $\mathtt {S}^{\prime }_{k_{-1}}$ is placed horizontally to the upper-right of $\mathtt {S}^{\prime }_{k_0}$ so that they are connected.

Figure 4 illustrates the above procedure.

Figure 4 The construction of $\mathtt {rd}(\underline {{\boldsymbol {\unicode{x3b1} }}}_{(1)})$ and $\mathtt {rd}(\underline {{\boldsymbol {\unicode{x3b1} }}}_{(2)})$ when $\alpha = (1,3,2,1)$ .

Now, let us construct $\partial ^1: \mathbf {P}_{\underline {{\boldsymbol {\unicode{x3b1} }}}}\rightarrow \boldsymbol {I}$ . Choose any tableau T in $\mathrm {SRT}(\underline {{\boldsymbol {\unicode{x3b1} }}})$ . Recall that $\mathbf {w}(T)$ is the word obtained by reading the entries of T from left to right, starting with the bottom row. Let $\mathbf {w}(T) = w_1 w_2 \cdots w_n$ . For each $1 \leq j \leq m$ , we consider the subword $\mathbf {w}_{T;j}$ of $\mathbf {w}(T)$ defined by

(4.3) $$ \begin{align} \mathbf{w}_{T;j} := w_{u_1(j)} w_{u_2(j)} \cdots w_{u_{l_j}(j)}, \end{align} $$

where the subscripts $u_i(j)$ s are defined via the following recursion:

$$ \begin{align*} &u_1(j) = \sum_{1 \le r \le j} (\alpha_{k_r}-1),\\ &u_{i+1}(j) = \min \{u_i(j) < u \le n - \ell(\alpha) \mid w_u < w_{u_i(j)} \} \quad (i \ge 1), \text{ and }\\ &l_j := \max \{i \mid u_i(j) < \infty \}. \end{align*} $$

In the second identity, whenever $\{u_i(j) < u \le n - \ell (\alpha ) \mid w_u < w_{u_i(j)} \}=\emptyset $ , we set $u_{i+1}(j):=\infty $ . Henceforth we simply write $u_i$ s for $u_i(j)$ s and thus $\mathbf {w}_{T;j}= w_{u_1} w_{u_2} \cdots w_{u_{l_j}}$ . Given an arbitrary word w, we use $\textsf {end}(w)$ to denote the last letter of w. With the notations above, we introduce the following two sets:

$$ \begin{align*} \begin{aligned} \mathtt{A}_{T;j} &: = \{y \in T(\mathtt{S}_{k_0}) \mid y> \textsf{end}(\mathbf{w}_{T;j})\}, \\ \mathcal{P}(\mathtt{A}_{T;j}) &: = \left\{ A \subseteq \mathtt{A}_{T;j} \mid |A| = \ell(\alpha)-k_j+1 \right\}. \end{aligned} \end{align*} $$

For $A \in \mathcal {P}(\mathtt {A}_{T;j})$ , we define $\tau _{T;j;A}$ to be an SRT of shape $\underline {{\boldsymbol {\unicode{x3b1} }}}_{(j)}$ that is uniquely determined by the following conditions:

1. (i) $\tau _{T;j;A}(\mathtt {S}^{\prime }_{k_{-1}}) = T(\mathtt {S}_{k_0}) \setminus A$ ,

2. (ii) $\tau _{T;j;A}(\mathtt {S}^{\prime }_{k_0}) =\{\textsf {end}(\mathbf {w}_{T;j})\} \cup A$ ,

3. (iii) $\tau _{T;j;A}(\mathtt {S}^{\prime }_{k_r}) = T(\mathtt {S}_{k_r})$ for $1 \leq r < j$ ,

4. (iv) $\tau _{T;j;A}(\mathtt {S}^{\prime }_{k_j}) = T(\mathtt {S}_{k_j}) \setminus \{w_{u_1} \}$ , and

5. (v) for $j < r \leq m$ , $\tau _{T;j;A}(\mathtt {S}^{\prime }_{k_r})$ is obtained from $T(\mathtt {S}_{k_r})$ by substituting $w_{u_{i}}$ with $w_{u_{i-1}}$ for $w_{u_{i}}$ s ( $1 < i \le l_j$ ) contained in $T(\mathtt {S}_{k_r})$ .

We next explain the notion of the signature $\mathrm {sgn}(A)$ of A. Enumerate the elements in $\mathtt {A}_{T;j}$ in the increasing order

$$\begin{align*}a_1 < a_2 < \cdots < a_{|\mathtt{A}_{T;j}|}. \end{align*}$$

Then let $A^1_{T;j}$ be the set of the consecutive $(\ell (\alpha )-k_j+1)$ elements starting from the rightmost and moving to the left, precisely,

$$ \begin{align*} A^1_{T;j} = \{a_{|\mathtt{A}_{T;j}|-\ell(\alpha)+k_j},a_{|\mathtt{A}_{T;j}|-\ell(\alpha)+k_j+1}, \ldots,a_{|\mathtt{A}_{T;j}|} \}. \end{align*} $$

There is a natural right $\Sigma _{|\mathtt {A}_{T;j}|}$ -action on $\mathtt {A}_{T;j}$ given by

(4.4) $$ \begin{align} a_i \cdot \omega = a_{\omega^{-1}(i)} \text{ for } 1 \le i \le |\mathtt{A}_{T;j}| \text{ and }\omega \in \Sigma_{|\mathtt{A}_{T;j}|}. \end{align} $$

We define $\mathrm {sgn}(A) := (-1)^{\ell (\omega ^1)}$ , where $\omega ^1$ is any minimal length permutation in $\{\omega \in \Sigma _{|\mathtt {A}_{T;j}|} \mid A = A^1_{T;j} \cdot \omega \}$ .

For each $1 \leq j \leq m$ , set

$$ \begin{align*} {\boldsymbol{\tau}}_{T;j}:= \displaystyle \sum_{A \in \mathcal{P}(\mathtt{A}_{T;j})} \mathrm{sgn}(A) \tau_{T;j;A}, \end{align*} $$

where the summation in the right-hand side is zero in the case where $\mathcal {P}(\mathtt {A}_{T;j})=\emptyset $ . Finally, we define a $\mathbb C$ -linear map

$$ \begin{align*} \partial^1: \mathbf{P}_{\underline{{\boldsymbol{\unicode{x3b1}}}}}\rightarrow \boldsymbol{I}, \quad T \mapsto \sum_{1 \leq j \leq m} {\boldsymbol{\tau}}_{T;j} \end{align*} $$

with $\boldsymbol {I}$ in equation (4.2).

Example 4.4. Let $\alpha = (2,1,2,3) \models 8$ . Then $\mathcal {K}(\alpha ) = \{0,1,3,4\}$ and $\underline {{\boldsymbol {\unicode{x3b1} }}} = (1) \oplus (1) \oplus (3,1^3)$ . By definition, we get

$$ \begin{align*} \underline{{\boldsymbol{\unicode{x3b1}}}}_{(1)} & = (1) \oplus (3,1^4), \\ \underline{{\boldsymbol{\unicode{x3b1}}}}_{(2)} & = (1) \oplus (3,1^2) \oplus (1^2), \\ \underline{{\boldsymbol{\unicode{x3b1}}}}_{(3)} & = (1) \oplus (1) \oplus (2^2,1^2). \end{align*} $$

(a) Let

. Then one sees that

$$ \begin{align*} \begin{array}{llll} \mathbf{w}_{T;1} = 6 \ 2 \quad & \quad \textsf{end}(\mathbf{w}_{T;1}) = 2 \quad & \mathtt{A}_{T;1} = \{3, 7,8\} \quad & \quad \mathcal{P}(\mathtt{A}_{T;1}) = \emptyset,\\ \mathbf{w}_{T;2} = 2 & \quad \textsf{end}(\mathbf{w}_{T;2}) = 2 & \mathtt{A}_{T;2} = \{3,7,8 \} & \quad \mathcal{P}(\mathtt{A}_{T;2}) = \{\{3,7\},\{3,8\},\{7,8\}\}, \\ \mathbf{w}_{T;3} = 4 & \quad \textsf{end}(\mathbf{w}_{T;3}) = 4 & \mathtt{A}_{T;3} = \{7,8 \} & \quad \mathcal{P}(\mathtt{A}_{T;3}) = \{\{7\},\{8\} \}. \end{array} \end{align*} $$

Since

it follows that

$$ \begin{align*} {\boldsymbol{\tau}}_{T;1} = 0 \qquad {\boldsymbol{\tau}}_{T;2} = \tau_{T;2;\{3,7\}} - \tau_{T;2;\{3,8\}} + \tau_{T;2;\{7,8\}} \qquad {\boldsymbol{\tau}}_{T;3} = - \tau_{T;3;\{7\}} + \tau_{T;3;\{8\}}. \end{align*} $$

Therefore,

$$ \begin{align*} \partial^1(T) = (\tau_{T;2;\{3,7\}} - \tau_{T;2;\{3,8\}} + \tau_{T;2;\{7,8\}}) + (- \tau_{T;3;\{7\}} + \tau_{T;3;\{8\}}). \end{align*} $$

(b) Note that

$$ \begin{align*} [\underline{{\boldsymbol{\unicode{x3b1}}}}_{(1)}] & = \left\{(1,3,1^4), (4,1^4)\right\}, \\ [\underline{{\boldsymbol{\unicode{x3b1}}}}_{(2)}] & = \left\{(1,3,1^4),(1,3,1,2,1),(4,1^4),(4,1,2,1)\right\}, \\ [\underline{{\boldsymbol{\unicode{x3b1}}}}_{(3)}] & = \left\{(1^2,2^2,1^2),(1,3,2,1^2),(2^3,1^2),(4,2,1^2) \right\}. \end{align*} $$

Theorem 4.3(d) implies that

$$ \begin{align*} \dim \mathrm{Ext}^1_{H_n(0)}(\mathbf{F}_{\beta},\mathcal{V}_{\alpha}) = \begin{cases} 1 & \text{ if } \beta^{\mathrm{r}} \in \mathcal{L}(\alpha) \setminus \{(1, 3, 1^4),(4, 1^4)\}, \\ 2 & \text{ if } \beta^{\mathrm{r}} \in \{(1,3,1^4), (4, 1^4)\}, \\ 0 & \text{ otherwise.} \end{cases} \end{align*} $$

5 $\mathrm {Ext}^i_{H_n(0)}(\mathcal {V}_{\alpha }, \mathcal {V}_{\beta })$ with $i=0,1$

In the previous sections, we computed $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$ and $\mathrm {Ext}^1_{H_n(0)}(\mathbf {F}_{\beta }, \mathcal {V}_{\alpha })$ . In this section, we focus on $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathcal {V}_{\beta })$ and $\mathrm {Ext}^0_{H_n(0)}(\mathcal {V}_{\alpha }, \mathcal {V}_{\beta })\, (=\mathrm {Hom}_{H_n(0)}(\mathcal {V}_{\alpha }, \mathcal {V}_{\beta }))$ .

Let $M, N$ be finite-dimensional $H_n(0)$ -modules. Given a short exact sequence

with $(P_0,\pi )$ a projective cover of M, it is well known that

$$ \begin{align*} \mathrm{Ext}_{H_n(0)}^1(M,N) \cong \frac{\mathrm{Hom}_{H_n(0)}(\Omega(M),N)} {\mathrm{Im} \,\iota^{\ast}}, \end{align*} $$

where $\iota ^{\ast}\!:\mathrm {Hom}_{H_n(0)}(P_0, N) \to \mathrm {Hom}_{H_n(0)}(\Omega (M),N)$ is given by composition with $\iota $ . The kernel of $\iota ^{\ast }$ equals

$$\begin{align*}\{f\in \mathrm{Hom}_{H_n(0)}(P_0,N) \mid f|_{\Omega(M)}=0\}, \end{align*}$$

and therefore

(5.1) $$ \begin{align} \ker(\iota^{\ast}) \cong \mathrm{Hom}_{H_n(0)}(P_0/\Omega(M),N)\cong \mathrm{Hom}_{H_n(0)}(M,N). \end{align} $$

This says that $\mathrm {Ext}_{H_n(0)}^1(M,N)=0$ if and only if, as $\mathbb C$ -vector spaces,

(5.2) $$ \begin{align} \mathrm{Hom}_{H_n(0)}(P_0, N) \cong \mathrm{Hom}_{H_n(0)}(\Omega(M), N) \oplus \mathrm{Hom}_{H_n(0)}(M,N). \end{align} $$

Whenever M is essentially rigid, one has that $\mathrm {Hom}_{H_n(0)}(P_0, M) \cong \mathrm {End}_{H_n(0)}(M).$ Typical examples of essentially rigid $H_n(0)$ -modules are simple modules and projective modules. The syzygy and cosyzygy modules of a rigid module are also rigid since $\mathrm {Ext}_{H_n(0)}^1(M,N)=\mathrm {Ext}_{H_n(0)}^1(\Omega (M),\Omega (N))$ and $M \cong \Omega \Omega ^{-1}(M) \oplus (\mathrm {projective})$ (for example, see [Reference Benson3]).

Let us use $\le _l$ to represent the lexicographic order on compositions of n. Using the results in the preceding sections, we derive some interesting results on $\mathrm {Ext}_{H_n(0)}^1(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ . To do this, we need the following lemmas.

We now state the main result of this section.

Proof. (a) Due to Theorem 3.3, there is a projective resolution of $\mathcal {V}_{\alpha }$ of the form

Hence, for the assertion, it suffices to show that

$$\begin{align*}\mathrm{Hom}_{H_n(0)}\left(\bigoplus_{i \in \mathcal{I}(\alpha)} \mathbf{P}_{{\boldsymbol{\unicode{x3b1}}}^{(i)}},\mathcal{V}_{\beta} \right)=0. \end{align*}$$

Observe that

$$ \begin{align*} \dim\mathrm{Hom}_{H_n(0)}\left(\bigoplus_{i \in \mathcal{I}(\alpha)} \mathbf{P}_{{\boldsymbol{\unicode{x3b1}}}^{(i)}},\mathcal{V}_{\beta} \right) &=\sum_{ \gamma \in \mathcal{J}(\alpha)}\dim\mathrm{Hom}_{H_n(0)}\left(\mathbf{P}_{\gamma},\mathcal{V}_{\beta} \right)\\ &=\sum_{ \gamma \in \mathcal{J}(\alpha)} [\mathcal{V}_{\beta}:\mathbf{F}_{\gamma}] \quad \text{ (by Lemma \hyperlink{lemma1}{5.2})}. \end{align*} $$

Here, $[\mathcal {V}_{\beta }:\mathbf {F}_{\gamma }]$ denotes the multiplicity of $\mathbf {F}_{\gamma }$ as a composition factor of $\mathcal {V}_{\beta }$ and thus equals the coefficient of $F_{\gamma }$ in the expansion of $\mathfrak {S}^*_{\beta }$ into fundamental quasisymmetric functions. From Lemma 5.3, it follows that this coefficient vanishes unless $\beta \ge _l \gamma $ . Since $\alpha <_l \gamma $ for all $\gamma \in \mathcal {J}(\alpha )$ , the assumption $\beta \le _l \alpha $ yields the desired result.

(b) Combining equation (5.2) with (a) yields that

$$ \begin{align*} \mathrm{Hom}_{H_n(0)}(\mathbf{P}_{\alpha}, \mathcal{V}_{\beta})\cong \mathrm{Hom}_{H_n(0)}(\Omega(\mathcal{V}_{\alpha}), \mathcal{V}_{\beta}) \oplus \mathrm{Hom}_{H_n(0)}(\mathcal{V}_{\alpha},\mathcal{V}_{\beta}). \end{align*} $$

But by Lemma 5.2 and Lemma 5.3, we see that

$$ \begin{align*} \dim\mathrm{Hom}_{H_n(0)}(\mathbf{P}_{\alpha}, \mathcal{V}_{\beta})=L_{\beta, \alpha}= \begin{cases} 1 & \text{ if } \beta =\alpha\\ 0 & \text{ otherwise.} \end{cases} \end{align*} $$

This justifies the assertion since $\dim \mathrm {End}_{H_n(0)}( \mathcal {V}_{\alpha })\ge 1$ .

(c) Let $f:\mathbf {P}_{\alpha } \to M$ be a surjective $H_n(0)$ -module homomorphism. Then

$$\begin{align*}\mathrm{End}_{H_n(0)}(M)\cong \mathrm{Hom}_{H_n(0)}(\mathbf{P}_{\alpha}/\ker(f), M), \end{align*}$$

and therefore

$$\begin{align*}1\le \dim\mathrm{End}_{H_n(0)}(M)\le \dim\mathrm{Hom}_{H_n(0)}(\mathbf{P}_{\alpha}, M)=[M: \mathbf{F}_{\alpha}]. \end{align*}$$

Now the assertion follows from the inequality $[M: \mathbf {F}_{\alpha }]\le [\mathcal {V}_{\alpha }: \mathbf {F}_{\alpha }]=L_{\alpha , \alpha }=1$ .

Theorem 5.4 (b) is no longer valid unless $\beta \le _l \alpha $ . In view of $\mathcal {V}_{\alpha } \cong \mathbf {P}_{\alpha } / \Omega (\mathcal {V}_{\alpha })$ , one can view $\mathrm {Hom}_{H_n(0)}(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ as the $\mathbb C$ -vector space consisting of $H_n(0)$ -module homomorphisms from $\mathbf {P}_{\alpha }$ to $\mathcal {V}_{\beta }$ that vanish on $\Omega (\mathcal {V}_{\alpha })$ . Therefore, to understand $\mathrm {Hom}_{H_n(0)}(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ , it is indispensable to understand $\mathrm {Hom}_{H_n(0)}(\mathbf {P}_{\alpha }, \mathcal {V}_{\beta })$ first. To do this, let us fix a linear extension $\preccurlyeq _{L}^{\mathrm {r}}$ of the partial order $\preccurlyeq ^{\mathrm {r}}$ on $\mathrm {SIT}(\beta )$ given by

$$ \begin{align*} \tau' \preccurlyeq^{\mathrm{r}}\tau \quad \text{if and only if} \quad \tau' = \pi_{\gamma} \cdot \tau \text{ for some } \gamma\in\Sigma_n. \end{align*} $$

Given $f\in \mathrm {Hom}_{H_n(0)}(\mathbf {P}_{\alpha },\mathcal {V}_{\beta })$ , let $f(T_{\alpha })=\sum _{\mathscr {T} \in \mathrm {SIT}(\beta )}c_{f,\mathscr {T}}\mathscr {T}$ . We define ${\mathsf {Lead}}(f)$ to be the largest tableau in $\{\mathscr {T} \in \mathrm {SIT}(\beta ): c_{f,\mathscr {T}}\ne 0\}$ with respect to $\preccurlyeq _{L}^{\mathrm {r}}$ . When $f=0$ , ${\mathsf {Lead}}(f)$ is set to be $\emptyset $ .

Proof. (a) Observe that every homomorphism in $\mathrm {Hom}_{H_n(0)}(\mathbf {P}_{\alpha },\mathcal {V}_{\beta })$ is completely determined by the value at the source tableau $T_{\alpha }$ of $\mathbf {P}_{\alpha }$ . We claim that $\mathrm {Des}({\mathsf {Lead}}(f)) =\mathrm {set}(\alpha )$ for all nonzero $f\in \mathrm {Hom}_{H_n(0)}(\mathbf {P}_{\alpha },\mathcal {V}_{\beta })$ . To begin with, from the equalities $f(\pi _i \cdot T_{\alpha })=f(T_{\alpha })$ for all $i\notin \mathrm {Des}(T_{\alpha })=\mathrm {set}(\alpha )$ , we see that f satisfies the condition that $\mathrm {Des}({\mathsf {Lead}}(f)) \subseteq \mathrm {set}(\alpha )$ . Recall that we set $m_i:=\sum _{1\le k \le i}\alpha _i$ for all $1\le i \le \ell (\alpha )$ in Section 3. Suppose that there is an index j such that

$$\begin{align*}m_j \in \mathrm{set}(\alpha) \setminus \mathrm{Des}({\mathsf {Lead}}(f)). \end{align*}$$

Then

$$\begin{align*}m_{j-1}+1, m_{j-1}+2, \ldots, m_{j+1}-1 \in \mathrm{set}(\alpha) \setminus \mathrm{Des}({\mathsf {Lead}}(f)). \end{align*}$$

But this is absurd since

$$\begin{align*}\pi_{[m_{j-1}+1, m_{j+1}-\alpha_j]^{\mathrm{r}}} \cdots \pi_{[m_j-1,m_{j+1}-2]^{\mathrm{r}}}\pi_{[m_j, m_{j+1}-1]^{\mathrm{r}}} \cdot T_{\alpha} = 0, \end{align*}$$

whereas

$$\begin{align*}\pi_{[m_{j-1}+1, m_{j+1}-\alpha_j]^{\mathrm{r}}} \cdots \pi_{[m_j-1,m_{j+1}-2]^{\mathrm{r}}}\pi_{[m_j, m_{j+1}-1]^{\mathrm{r}}} \cdot {\mathsf {Lead}}(f) = {\mathsf {Lead}}(f). \end{align*}$$

So the claim is verified.

For each $U \in {\mathfrak B}$ , consider the $\mathbb C$ -vector space

$$ \begin{align*} H(U):=\{f\in \mathrm{Hom}_{H_n(0)}(\mathbf{P}_{\alpha},\mathcal{V}_{\beta}): {\mathsf {Lead}}(f) \preccurlyeq_{L}^{\mathrm{r}} U\}. \end{align*} $$

Write ${\mathfrak B}$ as $\{U_1\preccurlyeq _{L}^{\mathrm {r}}U_{2} \preccurlyeq _{L}^{\mathrm {r}} \cdots \preccurlyeq _{L}^{\mathrm {r}} U_{l-1} \preccurlyeq _{L}^{\mathrm {r}} U_{l}\}$ , where $l=|{\mathfrak B}|$ . For any $f,g \in H(U_i)$ , it holds that

$$\begin{align*}c_{g,{\mathsf {Lead}}(g)}f- c_{f,{\mathsf {Lead}}(f)}g \in H(U_{i-1}) \end{align*}$$

with $H(U_0):=0$ . This implies that $\dim H(U_i)/H(U_{i-1}) \le 1$ for all $1\le i \le l$ .

Combining these inequalities with the equality $\dim \mathrm {Hom}_{H_n(0)}\left (\mathbf {P}_{\alpha },\mathcal {V}_{\beta } \right )=|{\mathfrak B}|$ , we deduce that, for each $U\in {\mathfrak B}$ , there exists a unique $f_U\in \mathrm {Hom}_{H_n(0)}(\mathbf {P}_{\alpha },\mathcal {V}_{\beta })$ with the desired property.

(b) By (a), one sees that $\{f_U: U\in {\mathfrak B}\}$ forms a basis for $\mathrm {Hom}_{H_n(0)}\left (\mathbf {P}_{\alpha },\mathcal {V}_{\beta } \right )$ . Since $\mathrm {Hom}_{H_n(0)}(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ is isomorphic to the $\mathbb C$ -vector space consisting of $H_n(0)$ -module homomorphisms from $\mathbf {P}_{\alpha }$ to $\mathcal {V}_{\beta }$ which vanish on $\Omega (\mathcal {V}_{\alpha })$ , our assertion follows from Lemma 6.2, which says that $\{\pi _{[m_{i-1} + 1, m_i]} \cdot T_{\alpha } \,: \, i \in \mathcal {I}(\alpha )\}$ is a generating set of $\Omega (\mathcal {V}_{\alpha })$ .

Example 5.7. (a) Let $\alpha =(1,1,2,1)$ and $\beta =(1,2,2)$ . Then

and

Note that $\mathcal {I}(\alpha )=\{2\}$ and $m_1=1, m_2=2$ . Since $\pi _{2} \cdot f_U(T_{\alpha })=0$ , it follows that $\mathrm {Hom}_{H_n(0)}(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ is $1$ -dimensional.

(b) Let $\alpha =(1,1,3,2)$ and $\beta =(2,3,2)$ . Then

and $f_{U_i}(T_{\alpha })=U_i$ for $i=1,2,3$ . Note that $\mathcal {I}(\alpha )=\{2,3\}$ and $m_1=1, m_2=2, m_3 = 5$ . Since $\pi _{2} \cdot f_{U_i}(T_{\alpha }) =0$ for all $1\le i \le 3$ and

it follows that $\mathrm {Hom}_{H_n(0)}(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ is $1$ -dimensional.

We end up with an interesting consequence of Theorem 4.3, where we successfully compute $\mathrm {Ext}_{H_n(0)}^1(\mathbf {F}_{\beta },\mathcal {V}_{\alpha })$ by constructing an injective hull of $\Omega ^{-1}(\mathcal {V}_{\alpha })$ . To compute it in a different way, let us consider a short exact sequence

Here, $\iota $ is the natural injection. Then we have

(5.3) $$ \begin{align} \mathrm{Ext}_{H_n(0)}^1(\mathbf{F}_{\beta},\mathcal{V}_{\alpha}) \cong \frac{\mathrm{Hom}_{H_n(0)}\left({\mathrm{rad}}(\mathbf{P}_{\beta}) ,\mathcal{V}_{\alpha} \right)} {\mathrm{Im}\, \iota^{\ast}}, \end{align} $$

where $\iota ^{\ast }: \mathrm {Hom}_{H_n(0)}( \mathbf {P}_{\beta } ,\mathcal {V}_{\alpha } )\rightarrow \mathrm {Hom}_{H_n(0)}({\mathrm {rad}}(\mathbf {P}_{\beta }),\mathcal {V}_{\alpha } ) $ is given by composition by with $\iota $ . By equation (5.1), one has that

$$ \begin{align*} \dim \mathrm{Im}\, \iota^{\ast} &=\dim \mathrm{Hom}_{H_n(0)}\left(\mathbf{P}_{\beta},\mathcal{V}_{\alpha} \right)-\dim \mathrm{Hom}_{H_n(0)}\left(\mathbf{F}_{\beta},\mathcal{V}_{\alpha} \right)\\ &=[\mathcal{V}_{\alpha}:\mathbf{F}_{\beta}]-[\mathrm{soc}(\mathcal{V}_{\alpha}):\mathbf{F}_{\beta}]\\ &=L_{\alpha, \beta}-[[\underline{{\boldsymbol{\unicode{x3b1}}}}]:\beta^{\mathrm{r}}] \quad \text{(by Lemma 5.3 and Theorem 4.1)}, \end{align*} $$

where $[[\underline {{\boldsymbol {\unicode{x3b1} }}}]:\beta ^{\mathrm {r}}]$ is the multiplicity of $\beta ^{\mathrm {r}} \in [\underline {{\boldsymbol {\unicode{x3b1} }}}]$ . Comparing Theorem 4.3 with equation (5.3) yields the following result.

Corollary 5.8. Let $\alpha , \beta $ be compositions of n. Then we have

$$ \begin{align*} \dim \mathrm{Hom}_{H_n(0)}\left({\mathrm{rad}}(\mathbf{P}_{\beta}) ,\mathcal{V}_{\alpha} \right) =L_{\alpha, \beta}-[[\underline{{\boldsymbol{\unicode{x3b1}}}}]:\beta^{\mathrm{r}}]+ [\mathcal{L}(\alpha):\beta^{\mathrm{r}}]. \end{align*} $$

6 Proof of Theorems

6.1 Proof of Theorem 3.3

We first prove that $\Omega (\mathcal {V}_{\alpha })$ is generated by $\{T^{(i)}_{\alpha } \mid i \in \mathcal {I}(\alpha )\}$ . By the definition of $\Phi $ , one can easily derive that

$$ \begin{align*} \Omega(\mathcal{V}_{\alpha}) = \mathbb C \{T \in \mathrm{SRT}(\alpha) \mid T_p^1> T_{p+1}^1 \text{ for some } 1\leq p < \ell(\alpha)\}. \end{align*} $$

Given $\sigma \in \Sigma _n$ , let

$$\begin{align*}\mathrm{Des}_L(\sigma):= \{i \in [n-1] \mid \ell(s_i \sigma) < \ell(\sigma)\} \ \ \text{and} \ \ \mathrm{Des}_R(\sigma):= \{i \in [n-1] \mid \ell(\sigma s_i) < \ell(\sigma)\}. \end{align*}$$

The left weak Bruhat order $\preceq _L$ on $\Sigma _n$ is the partial order on $\Sigma _n$ whose covering relation $\preceq _L^c$ is defined as follows: $\sigma \preceq _L^c s_i \sigma $ if and only if $i \notin \mathrm {Des}_L(\sigma )$ . It should be remarked that a word of length n can be confused with a permutation in $\Sigma _n$ if each of $1,2,\ldots , n$ appears in it exactly once.

The following lemma plays a key role in proving Lemma 6.2.

Lemma 6.1 ([Reference Björner and Brenti8, Proposition 3.1.2 (vi)]).

Suppose that $i \in \mathrm {Des}_R(\sigma ) \cap \mathrm {Des}_R(\rho )$ . Then $\sigma \preceq _L \rho $ if and only if $\sigma s_i \preceq _L \rho s_i$ .

Lemma 6.2. For each $i \in \mathcal {I}(\alpha )$ , $H_n(0) \cdot T^{(i)}_{\alpha } = \mathbb C\{T \in \mathrm {SRT}(\alpha ) \mid T_i^1> T_{i+1}^1 \}$ . Thus, $\Omega (\mathcal {V}_{\alpha }) = \sum _{i\in \mathcal {I}(\alpha )} H_n(0) \cdot T^{(i)}_{\alpha }$ .

Proof. For simplicity, let $\mathrm {SRT}(\alpha )^{(i)}$ be the set of $\mathrm {SRT}$ x of shape $\alpha $ such that the topmost entry in the ith column is greater than that in the $(i+1)$ st column.

We first show that $H_n(0) \cdot T^{(i)}_{\alpha }$ is included in the $\mathbb C$ -span of $\mathrm {SRT}(\alpha )^{(i)}$ , equivalently $\pi _{\sigma } \cdot T^{(i)}_{\alpha } \in \mathrm {SRT}(\alpha )^{(i)} \cup \{0\}$ for all $\sigma \in \Sigma _n$ . Suppose that there exists $\sigma \in \Sigma _n$ such that $\pi _{\sigma } \cdot T^{(i)}_{\alpha } \neq 0$ and $\pi _{\sigma } \cdot T^{(i)}_{\alpha } \notin \mathrm {SRT}(\alpha )^{(i)}$ . Let $\sigma _0$ be such a permutation with minimal length and j a left descent of $\sigma _0$ . By the minimality of $\sigma _0$ , we have $\pi _{s_j \sigma _0} \cdot T^{(i)}_{\alpha } \in \mathrm {SRT}(\alpha )^{(i)}$ , and therefore

$$\begin{align*}(\pi_{s_j \sigma_0} \cdot T^{(i)}_{\alpha})^1_{i}> (\pi_{s_j \sigma_0} \cdot T^{(i)}_{\alpha})^1_{i+1}.\end{align*}$$

By the definition of the $\pi _j$ -action on $\mathrm {SRT}(\alpha )$ , we have

$$\begin{align*}( \pi_j \cdot (\pi_{s_j \sigma_0} \cdot T^{(i)}_{\alpha}) )^1_i> ( \pi_j \cdot (\pi_{s_j \sigma_0} \cdot T^{(i)}_{\alpha}) )^1_{i+1}.\end{align*}$$

However, since