Quantum Mass Production Theorems
William KretschmerUniversity of Texas at Austin.  Email:
.  Supported by an NDSEG Fellowship.

We prove that for any $n$-qubit unitary transformation $U$ and for any $r = 2^{o(n / \log n)}$, there exists a quantum circuit to implement $U^{\otimes r}$ with at most $O(4^n)$ gates. This asymptotically equals the number of gates needed to implement just a single copy of a worst-case $U$. We also establish analogous results for quantum states and diagonal unitary transformations. Our techniques are based on the work of Uhlig [Math. Notes 1974], who proved a similar mass production theorem for Boolean functions.

§ INTRODUCTION

If a computational task requires $c$ resources, then common sense dictates that repeating the same task $r$ times should require roughly $c \cdot r$ resources. In many settings,
including query complexity <cit.> and communication complexity <cit.>, this intuition can be made rigorous: such results are known as direct sum theorems.

Closely related are direct product theorems,

which show that, with a fixed computational budget, the probability of successfully performing $r$ independent tasks decays in $r$. We recommend <cit.> for a good overview of the topic.

Nevertheless, direct sum and direct product theorems are not universal. Some computational settings exhibit a “mass production” phenomenon, in which the cost of performing the same task many times in parallel does not scale linearly with the number of repetitions. A well-known example <cit.> is based on the circuit complexity of matrix-vector multiplication. For a matrix $M \in \{0,1\}^{n \times n}$, define $f_M: \{0,1\}^n \to \{0,1\}^n$ by $f_M(v) = Mv$, where addition and multiplication are taken mod $2$. Then a simple counting argument implies that for most $M$, the complexity of implementing $f_M$ via a Boolean circuit is at least $\Omega(n^2 / \log n)$, as measured by the number of $2$-bit $\AND$, $\OR$, and $\NOT$ gates. Yet, by observing that $f_M^n$ (i.e. $f_M$ repeated $n$ times) is simply a matrix-matrix multiplication, we find that the cost of implementing $f_M^n$ is only $O(n^\omega)$, where $\omega < 2.38$ is the exponent of matrix multiplication <cit.>—substantially less than the naive bound of $O(n^3)$.

One might be left with the impression that such mass production phenomena can only occur for extremely special functions, like matrix multiplication, that have a particular algebraic or combinatorial structure. Remarkably, this intuition fails dramatically in the setting of Boolean circuit complexity. A theorem of Uhlig <cit.> shows that for any Boolean function $f: \{0,1\}^n \to \{0,1\}$ and for any $r = 2^{o(n / \log n)}$, there exists a Boolean circuit implementing $f^r$ with at most $O\left(\frac{2^n}{n}\right)$ gates. Asymptotically, this equals the number of gates needed to evaluate a worst-case $f$ on a single input, by the well-known counting argument of Shannon <cit.>. In fact, Uhlig even showed that the leading constant in the big-$O$ does not increase with $r$, and hence arbitrary Boolean functions can be mass produced with essentially no overhead.

In this work, we consider the natural question of whether a similar mass production phenomenon holds for quantum circuit complexity. Our question is well-motivated by recent works demonstrating that for certain learning tasks, algorithms with access to many copies of a quantum state on a quantum memory can be exponentially more powerful than algorithms that have access only to single copies of the state at a time <cit.>. Indeed, these results suggest that optimizing the complexity of mass producing quantum states and processes could have valuable applications. We also view our question as interesting from a purely theoretical perspective, especially considering that Uhlig's theorem for classical functions has recently found complexity-theoretic applications in characterizing the minimum circuit size problem <cit.>.

For simplicity, we consider quantum circuit complexity in the setting of qubit quantum circuits, using the universal gate set of arbitrary single-qubit gates plus $\CNOT$ gates with all-to-all connectivity. We also allow ancilla qubits initialized to $\ket{0}$, so long as they are reset to $\ket{0}$ at the end of the computation. We measure circuit complexity in terms of the $\CNOT$ count. This measure is justified by the fact that multiple-qubit gates are more error-prone and expensive to implement than single-qubit gates, and also by the observation that the number of single-qubit gates is related to the $\CNOT$ count by at most a factor of $4$ in any irredundant circuit.

In analogy with Uhlig's theorem <cit.>, our main result establishes mass production theorems for both quantum states and unitary transformations.

Let $\ket{\psi}$ be an $n$-qubit quantum state, and let $r = 2^{o(n / \log n)}$. Then there exists a quantum circuit with at most $(1 + o(1))2^n$ $\CNOT$ gates to prepare $\ket{\psi}^{\otimes r}$.

Let $U$ be an $n$-qubit unitary transformation, and let $r = 2^{o(n / \log n)}$. Then there exists a quantum circuit with at most $(5/2 + o(1))4^n$ $\CNOT$ gates to implement $U^{\otimes r}$.

Note that the factor $2^n$ (respectively, $4^n$), in <Ref> (respectively, <Ref>) is optimal, because it asymptotically equals the number of $\CNOT$ gates needed to prepare a single copy of an arbitrary $n$-qubit state (respectively, to implement an arbitrary $n$-qubit unitary once),  up to a small multiplicative constant <cit.>. Above, we made the leading constants explicit only to illustrate that they are not too large, and thus to demonstrate that these theorems have some hope of becoming practical. We leave a full optimization of these constants and the factors hidden in the $o(1)$ to future work.

Our results build heavily on the simple proof of Uhlig's theorem given in <cit.>. Indeed, our main contribution is the observation that Uhlig's proof generalizes from Boolean functions to diagonal unitary matrices (<Ref>). From there, using well-known decompositions of quantum states and unitary transformations into diagonal gates <cit.>, we obtain the corresponding mass production theorems.

§ PRELIMINARIES

§.§ Basic Notation

We denote by $\ind\{p\}$ the function that evaluates to $1$ if proposition $p$ is true, and $0$ otherwise. If $\alpha$ is a complex number, we let $\alpha^*$ denote its complex conjugate. We denote by $\Units \coloneqq \{a + bi : |a|^2 + |b|^2 = 1\}$ the set of complex units. For a function $f: \{0,1\}^n \to \Units$, denote by $\bar{f}: \{0,1\}^{n+1} \to \Units$ the function defined by $\bar{f}(x, c) = f(x)^{1 - 2c}$. We freely identify a function $f: \{0,1\}^n \to \Units$ with the corresponding diagonal unitary transformation $U$ that acts as $U\ket{x} = f(x)\ket{x}$ on basis states $x \in \{0,1\}^n$.

We use standard notation for quantum circuits, including $\CNOT$, Toffoli, and Fredkin gates. We also borrow a large amount of notation and terminology from <cit.>, as we detail further below. We define the $x$-, $y$-, and $z$-axis rotations by:

\begin{align*}
R_x(\theta) &= \begin{pmatrix}
\cos(\theta / 2) & i \sin(\theta / 2)\\
i \sin(\theta/2) & \cos(\theta / 2)
\end{pmatrix}\\
R_y(\theta) &= \begin{pmatrix}
\cos(\theta / 2) & \sin(\theta / 2)\\
- \sin(\theta/2) & \cos(\theta / 2)
\end{pmatrix}\\
R_z(\theta) &= \begin{pmatrix}
e^{-i \theta / 2} & 0\\
0 & e^{i \theta / 2}
\end{pmatrix}
\end{align*}

§.§ Multiplexors

A multiplexor with $s$ select qubits and $d$ data qubits is a block-diagonal $(s+d)$-qubit unitary transformation that preserves every computational basis state $\ket{x}$ on the select qubits. For brevity, we call such a unitary an $(s,d)$-multiplexor. An $(s,1)$-multiplexor in which all of the diagonal blocks are $R_z$ on the data qubit may alternatively be called a multiplexed $R_z$ (analogously for $R_x$ and $R_y$). Collectively, multiplexed $R_x$, $R_y$, and $R_z$ are called multiplexed rotations. Observe that an $(s,1)$-multiplexed $R_z$ is equivalent to a unitary implementing $\bar{f}$ for some $f: \{0,1\}^s \to \Units$.

We require the following basic fact about implementing multiplexed rotations:

Let $U$ be an $(n,1)$-multiplexed rotation. Then there exists a quantum circuit with at most $2^n$ $\CNOT$ gates to implement $U$.

§.§ Generic Gates

As in <cit.>, we use circuit diagrams containing generic gates. An equivalence of two circuit diagrams containing generic gates means that for any assignment of parameters to the generic gates on one side, there exists an assignment of parameters to the gates on the other side that makes the two circuits compute the same operator. We use the following notation for generic gates:

$\Qcircuit @C=1em @R=.7em { & {/} \qw & \gate{\phantom{U}} & \qw }$
A generic unitary gate.

$\Qcircuit @C=1em @R=.7em { & {/} \qw & \gate{\Delta} & \qw }$
A generic diagonal unitary gate.

$\Qcircuit @C=1em @R=.7em { & \gate{R_z} & \qw }$     An $R_z$ gate for some unspecified $\theta$. Conventions for
$R_x$ and $R_y$ are analogous.

$\Qcircuit @C=1em @R=.7em {
& {/} \qw & \controlu \qw & \qw\\
& {/} \qw & \gate{\phantom{U}} \qwx & \qw\\
}$     A generic multiplexor, with select qubits on the upper register and data qubits on the lower register

$\Qcircuit @C=1em @R=.7em {
& {/} \qw & \controlu \qw & \qw\\
& \qw & \gate{R_z} \qwx & \qw\\
}$     A multiplexed $R_z$. Conventions for
$R_x$ and $R_y$ are analogous.

§ DIAGONAL UNITARIES AND MULTIPLEXORS

We begin by generalizing the proof of Uhlig's theorem <cit.> to diagonal unitary matrices (or, more precisely, multiplexed $R_z$ gates).

Let $f: \{0,1\}^n \to \Units$ and let $r = 2^{o(n / \log n)}$. Then there exists a quantum circuit with at most $(1 + o(1))2^n$ $\CNOT$ gates to implement $\bar{f}^{\otimes r}$.

Without loss of generality, let $r = 2^t$ for some $t = o(n / \log n)$. Our proof proceeds by induction on $t$: for fixed $k$ (chosen later) and for every $n > k \cdot t$, we construct for each $f : \{0,1\}^n \to \Units$ a circuit $\mathcal{C}_{f,n,k,t}$ computing $\bar{f}^{\otimes 2^t}$. We proceed in order: first we construct $\mathcal{C}_{f,n,k,1}$ for every $n$ and $f$, then $\mathcal{C}_{f,n,k,2}$ for every $n$ and $f$, then $\mathcal{C}_{f,n,k,3
}$ for every $n$ and $f$, and so on. Ultimately, we show that there exists a universal constant $d$ such that the number of $\CNOT$ gates in $\mathcal{C}_{f,n,k,t}$, denoted $s_{n,k,t}$, satisfies the bound:

\begin{equation}
\label{eq:bound_s_nkt}
s_{n,k,t} \le \left(2^k + 1\right)^t \left(2^{n - tk} + 2^t dn \right).
\end{equation}

We begin by describing the construction of $\mathcal{C}_{f,n,k,1}$. For each $0 \le i \le 2^k - 1$, let $f_i: \{0,1\}^{n - k} \to \Units$ denote the restriction of $f$ obtained by fixing the first $k$ bits to the binary representation of $i$. For each $0 \le i \le 2^k$, define $g_i: \{0,1\}^{n - k} \to \Units$ by:

* $g_0 = f_0$.

* $g_\ell = f_{\ell - 1}^* f_\ell$ if $1 \le \ell \le 2^k - 1$.

* $g_{2^k} = f_{2^k - 1}^*$.

Observe that

\begin{equation}
f_i = \prod_{\ell = 0}^{i} g_\ell = \prod_{\ell = i+1}^{2^k} g_\ell^*,\label{eq:f_i_decomp}
\end{equation}

The key idea in the remainder of the proof is to evaluate $\bar{f}$ on a pair of inputs $(x, y)$ using the two decompositions in (<ref>), one each for $x$ and $y$. Indeed, the following algorithm accomplishes this.

Evaluate $\bar{f}^{\otimes 2}$

$x, y \in \{0,1\}^n$, $c_x, c_y \in \{0,1\}$
$\bar{f}(x, c_x) \cdot \bar{f}(y, c_y)$

$\alpha \coloneqq 1$

(*[f]viewing $x,y$ as integers with highest order bits $x_1,y_1$)$x \le y$
$m \coloneqq x$; $c_m \coloneqq c_x$ *[r]set $m = \min\{x, y\}$, $M = \max\{x, y\}$

$M \coloneqq y$; $c_M \coloneqq c_y$

$m \coloneqq y$; $c_m \coloneqq c_y$

$M \coloneqq x$; $c_M \coloneqq c_x$

$0 \le \ell \le 2^k$
(*[f]$x_\range{i}{j}$ denotes bits $i$ through $j$ of $x$)$\ell \le m_\range{1}{k}$
Multiply $\alpha$ by $\bar{g}_\ell(m_\range{k+1}{n}, c_m)$

$\ell > M_\range{1}{k}$
Multiply $\alpha$ by $\bar{g}_\ell(M_\range{k+1}{n}, 1 - c_{M})$ *[r]note negation on $c_{M}$

Multiply $\alpha$ by $1$

Here, the $\ell \le m_\range{1}{k}$ clause corresponds to the multiplication $\prod_{\ell = 0}^{m_\range{1}{k}} g_\ell$, while the $\ell > M_\range{1}{k}$ clause corresponds to $\prod_{\ell = M_\range{1}{k}}^{2^n} g_\ell^*$. An equivalent reformulation of <Ref> is given below.

Evaluate $\bar{f}^{\otimes 2}$

$x, y \in \{0,1\}^n$, $c_x,c_y \in \{0,1\}$
$\bar{f}(x, c_x) \cdot \bar{f}(y, c_y)$

$\alpha \coloneqq 1$

$x \le y$
$m \coloneqq x$; $c_m \coloneqq c_x$

$M \coloneqq y$; $c_M \coloneqq c_y$

$m \coloneqq y$; $c_m \coloneqq c_y$

$M \coloneqq x$; $c_M \coloneqq c_x$

$0 \le \ell \le 2^k$
$a \coloneqq \ind \{\ell \le m_\range{1}{k}\}$ *[r]at most one of $a, b$ is nonzero

$b \coloneqq \ind \{\ell > M_\range{1}{k}\}$

$z \coloneqq a \cdot m_\range{k+1}{n} \oplus b \cdot M_\range{k+1}{n}$

$c \coloneqq a \cdot c_m \oplus b \cdot (1 - c_M)$

Multiply $\alpha$ by $\bar{g}_\ell(z, c)$

Multiply $\alpha$ by $g_\ell^*(0^{n-k})^{(1 - a) \cdot (1 - b)}$ *[r]undo added phase in case $a = b = 0$

<Ref> readily extends to a quantum circuit implementation. Define a pair of classical reversible circuits $\mathcal{A}_n$ and $\mathcal{B}_{n,k,\ell}$ whose input and output behavior are given in <Ref>. Using $\mathcal{A}_n$ and $\mathcal{B}_{n,k,\ell}$, via the same strategy as <Ref>, we obtain the quantum circuit $\mathcal{C}_{f,n,k,1}$ defined in <Ref> that implements $\bar{f}^{\otimes 2}$.

\[
\Qcircuit @C=1em @R=0.7em {
\lstick{x \in \{0,1\}^n}
& {/} \qw
& \multigate{2}{\mathcal{A}_n}
& \rstick{x} \qw
\\
\lstick{y \in \{0,1\}^n}
& {/} \qw
& \ghost{\mathcal{A}_n}
& \rstick{y} \qw
\\
\lstick{0}
& \qw
& \ghost{\mathcal{A}_n}
& \rstick{\ind\{x > y\}} \qw
\\
\]

\[
\Qcircuit @C=1em @R=0.7em {
\lstick{0}
& \qw
& \multigate{3}{\mathcal{B}_{n,k,\ell}}
& \rstick{\ind\{\ell \le m_\range{1}{k}\}} \qw
\\
\lstick{0}
& \qw
& \ghost{\mathcal{B}_{n,k,\ell}}
& \rstick{\ind\{\ell > M_\range{1}{k}\}} \qw
\\
\lstick{m \in \{0,1\}^n}
& {/} \qw
& \ghost{\mathcal{B}_{n,k,\ell}}
& \rstick{m} \qw
\\
\lstick{M \in \{0,1\}^n}
& {/} \qw
& \ghost{\mathcal{B}_{n,k,\ell}}
& \rstick{M} \qw
\\
\]

Inputs and outputs of reversible circuits $\mathcal{A}_n$ and $\mathcal{B}_{n,k,\ell}$.

\[
\Qcircuit @C=1em @R=1.2em @!R {
\lstick{\ket{0^{n-k}}}
& {/} \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \targ
& \targ
& \qw
& \qw
& \ustick{_{z}} \qw
& \multigate{1}{\bar{g}_\ell}
& \qw
& \qw
& \targ
& \targ
& \qw
& \qw
& \qw
& \qw
& \qw
\\
\lstick{\ket{0}}
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \targ
& \targ
& \ustick{_{c}} \qw
& \ghost{\bar{g}_\ell}
& \targ
& \targ
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
\\
\lstick{\ket{0}}
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \multigate{3}{\mathcal{B}_{n,k,\ell}}
& \ustick{_{a}} \qw
& \ctrl{-2}
& \qw
& \ctrl{-1}
& \qw
& \qw
& \multigate{1}{*}
& \qw
& \ctrl{-1}
& \qw
& \ctrl{-2}
& \multigate{3}{\mathcal{B}^\dagger_{n,k,\ell}}
& \qw
& \qw
& \qw
& \qw
\\
\lstick{\ket{0}}
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ghost{\mathcal{B}_{n,k,\ell}}
& \ustick{_{b}} \qw
& \qw
& \ctrl{-3}
& \qw
& \ctrl{-2}
& \qw
& \ghost{*}
& \ctrl{-2}
& \qw
& \ctrl{-3}
& \qw
& \ghost{\mathcal{B}^\dagger_{n,k,\ell}}
& \qw
& \qw
& \qw
& \qw
\\
\lstick{x}
& {/} \qw
& \multigate{2}{\mathcal{A}_n}
& \qw
& \qswap
& \qw
& \ustick{_{m}} \qw
& \qw
& \ghost{\mathcal{B}_{n,k,\ell}}
& \qw
& \ctrl{-4}
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ctrl{-4}
& \ghost{\mathcal{B}^\dagger_{n,k,\ell}}
& \qw
& \qswap
& \multigate{2}{\mathcal{A}_n^\dagger}
& \qw
\\
\lstick{y}
& {/} \qw
& \ghost{\mathcal{A}_n}
& \qw
& \qswap
& \qw
& \ustick{_{M}} \qw
& \qw
& \ghost{\mathcal{B}_{n,k,\ell}}
& \qw
& \qw
& \ctrl{-5}
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ctrl{-5}
& \qw
& \ghost{\mathcal{B}^\dagger_{n,k,\ell}}
& \qw
& \qswap
& \ghost{\mathcal{A}_n^\dagger}
& \qw
\\
\lstick{\ket{0}}
& \qw
& \ghost{\mathcal{A}_n}
& \ustick{_{x>y}} \qw
& \ctrl{-2}
& \ctrl{2}
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ctrl{2}
& \ctrl{-2}
& \ghost{\mathcal{A}_n^\dagger}
& \qw
\\
\lstick{c_x}
& \qw
& \qw
& \qw
& \qw
& \qswap
& \ustick{_{c_m}} \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ctrl{-6}
& \qw
& \qw
& \qw
& \qw
& \ctrl{-6}
& \qw
& \qw
& \qw
& \qswap
& \qw
& \qw
& \qw
\\
\lstick{c_y}
& \qw
& \qw
& \qw
& \qw
& \qswap
& \ustick{_{c_M}} \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ctrlo{-7}
& \qw
& \qw
& \ctrlo{-7}
& \qw
& \qw
& \qw
& \qw
& \qswap
& \qw
& \qw
& \qw
\\
&&&&&&&&&&&&&& \raisebox{1.2em}{Repeat for each $0 \le \ell \le 2^k$}
\gategroup{1}{9}{9}{21}{4.6em}{.}
\]

Circuit diagram of $\mathcal{C}_{f,n,k,1}$. The Toffoli gates with controls acting on the $x$ and $y$ registers are understood to be arrays of $n-k$ Toffoli gates between the corresponding qubits of the control and target registers. The gate marked $*$ adds a phase of $g_\ell^*(0^{n-k})$ if both qubits are $\ket{0}$ and otherwise does nothing. For convenience, several of the wires are labeled with the values they take on corresponding to variables in <Ref>.

By <Ref>, for every $\ell$, $\bar{g}_\ell$ can be implemented using at most $2^{n-k}$ $\CNOT$ gates, because $\bar{g}_\ell$ is equivalent to an $(n-k,1)$-multiplexed $R_z$. Moreover, it is easy to see that $\mathcal{A}_n$ and $\mathcal{B}_{n,k,\ell}$ can be implemented using at most $O(n)$ $\CNOT$ gates each, because comparison of two $n$-bit integers can be performed by a classical circuit of at most $O(n)$ gates. As a consequence, we conclude that there exists a constant $d$ such that:

\begin{equation}
\label{eq:bound_s_nk1}
s_{n,k,1} \le \left(2^k + 1\right) \left(2^{n - k} + dn \right).
\end{equation}

This is certainly less than the bound in (<ref>), so this establishes the base case of the induction proof.

Now we proceed to the induction step on $t$. Suppose that for every $n > k \cdot (t - 1)$, we have a circuit $\mathcal{C}_{f,n,k,t-1}$ computing $\bar{f}^{\otimes 2^{t-1}}$ with $\CNOT$ count bounded by

\begin{equation}
\label{eq:bound_s_nkt-1}
s_{n,k,t-1} \le \left(2^k + 1\right)^{t-1} \left(2^{n - (t-1)k} + 2^{t-1} dn \right).
\end{equation}

To construct $\mathcal{C}_{f,n,k,t}$, we start by first taking $2^{t-1}$ copies of $\mathcal{C}_{f,n,k,1}$. Then, for each $0 \le \ell \le 2^k$, we replace each of the $2^{t-1}$ sub-circuits that compute $\bar{g}_\ell$ with $\mathcal{C}_{g_\ell,n-k,k,t-1}$. Then, the number of gates in $\mathcal{C}_{f,n,k,t}$ is bounded by:

\begin{align*}
s_{n,k,t} &\le \left(2^k + 1\right) \left(s_{n-k,k,t-1} + 2^{t-1} dn \right)\\
&\le \left(2^k + 1\right) \left(\left(2^k + 1\right)^{t-1} \left(2^{n - k - (t-1)k} + 2^{t-1} d(n-k) \right) + 2^{t-1} dn \right)\\
&\le \left(2^k + 1\right)^t \left(2^{n - tk} + 2^{t-1} dn \right) + \left(2^k + 1\right)2^{t-1} dn\\
&\le \left(2^k + 1\right)^t \left(2^{n - tk} + 2^t dn \right),
\end{align*}

where the first line substitutes (<ref>) for the cost of the $\bar{g}_\ell$'s and otherwise uses the same bound as (<ref>) for the non-$\bar{g}_\ell$ gates, and the second line applies the induction hypothesis (<ref>). This establishes the induction step, and thus (<ref>) holds for every $n > k \cdot t$.

Choose $k = \lceil \log n \rceil$. Then:

\begin{align*}
s_{n,k,t} &\le \left(2^k + 1\right)^t \left(2^{n - tk} + 2^t dn \right)\\
&= 2^{kt} \left(1 + \frac{1}{2^k}\right)^t \left(2^{n - tk} + 2^t dn \right)\\
&\le 2^{kt} e^{t / 2^k} \left(2^{n - tk} + 2^t dn \right)\\
&\le 2^{kt} (1 + o(1)) \left(2^{n - tk} + 2^t dn \right)\\
&\le 2^{kt} (1 + o(1)) \left(2^{n - tk} + o\left(2^{n - tk}\right) \right)\\
&\le (1 + o(1))2^n,
\end{align*}

where we applied the exponential inequality in the third line, and used the assumption $t \le o(n / \log n)$ in the fourth and fifth lines. This proves the theorem.

<Ref> straightforwardly generalizes to arbitrary multiplexed rotations and multiplexors with a single data qubit, as below.

Let $U$ be an $(n,1)$-multiplexed rotation, and let $r = 2^{o(n / \log n)}$. Then there exists a quantum circuit with at most $(1 + o(1))2^n$ $\CNOT$ gates to implement $U^{\otimes r}$.

The $R_z$ case follows by observing that $\bar{f}$ is exactly an $(n,1)$-multiplexed $R_z$ in <Ref>. This also extends to multiplexed $R_x$ and $R_y$, because multiplexed $R_x$, $R_y$, and $R_z$ are equivalent up to conjugation by single-qubit unitaries on the data qubit. That is, there exist single-qubit unitaries $U$ and $V$ such that:

\[
\hspace{2.5em}
\Qcircuit @C=1em @R=0.7em {
& {/} \qw
& \controlu \qw
& \qw
& \ccteq{1}
&&& {/} \qw
& \qw
& \controlu \qw
& \qw
& \qw
& \ccteq{1}
&&& {/} \qw
& \qw
& \controlu \qw
& \qw
& \qw
\\
& \qw
& \gate{R_z} \qwx
& \qw
& \ccteqg
&&& \qw
& \gate{U}
& \gate{R_x} \qwx
& \gate{U^\dagger}
& \qw
& \ccteqg
&&& \qw
& \gate{V}
& \gate{R_y} \qwx
& \gate{V^\dagger}
& \qw
\]

Hence, the $\CNOT$ count is identical for multiplexed $R_x$ and $R_y$ as well.

Let $U$ be an $(n,1)$-multiplexor, and let $r = 2^{o(n / \log n)}$. Then there exists a quantum circuit with at most $(4 + o(1))2^n$ $\CNOT$ gates to implement $U^{\otimes r}$.

By <cit.>, an arbitrary $(n,1)$-multiplexor may be implemented via a product of $4$ $(n,1)$-multiplexed rotations, as below.

\[
\hspace{2.5em}
\Qcircuit @C=1em @R=0.7em {
\lstick{\ket{0}}
& \qw
& \qw
& \qw
& \ccteq{2}
&& \lstick{\ket{0}}& \qw
& \qw
& \qw
& \qw
& \qw
& \qw
& \ccteq{2}
&& \lstick{\ket{0}}& \qw
& \qw
& \qw
& \qw
& \gate{R_z}
& \qw
\\
& {/} \qw
& \controlu \qw
& \qw
& \ccteqg
&&& {/} \qw
& \controlu \qw
& \controlu \qw
& \controlu \qw
& \gate{\Delta} \qw
& \qw
& \ccteqg
&&& {/} \qw
& \controlu \qw
& \controlu \qw
& \controlu \qw
& \controlu \qwx[-1] \qw
& \qw
\\
& \qw
& \gate{\phantom{U}} \qwx
& \qw
& \ccteqg
&&& \qw
& \gate{R_z} \qwx
& \gate{R_y} \qwx
& \gate{R_z} \qwx
& \qw
& \qw
& \ccteqg
&&& \qw
& \gate{R_z} \qwx
& \gate{R_y} \qwx
& \gate{R_z} \qwx
& \qw
& \qw
\]

Applying <Ref> to each of the multiplexed rotations on the right side above completes the proof.

§ STATES AND GENERAL UNITARIES

We now prove the main results of this work that generalize the mass production theorems above to state preparation and unitary compilation. The proofs proceed via the techniques of <cit.>, by decomposing operators into multiplexors.

By <cit.>, for any $n$-qubit quantum state $\ket{\psi}$, there exists an $(n-1)$-qubit state $\ket{\varphi}$ such that $\ket{\psi}$ has the following decomposition.

\[
\Qcircuit @C=1em @R=.7em
& {/} \qw
& \controlu \qwx[1] \qw
& \controlu \qwx[1] \qw
& \rstick{\ket{\varphi}} \qw
\\
& \qw
& \gate{R_z}
& \gate{R_y}
& \rstick{\ket{0}}\qw
\inputgroupv{1}{2}{1em}{1em}{\ket{\psi}}
\]

Applying this decomposition recursively, we conclude that $\ket{\psi}$ can be prepared by a circuit consisting of a pair of $(\ell,1)$-multiplexed rotations for each $1 \le \ell \le n-1$, and a pair of single-qubit gates.

Apply <Ref> to the $(\ell,1)$-multiplexed rotations for each $\lceil n/2 \rceil \le \ell \le n-1$, and otherwise apply <Ref> $r$ times for each $1 \le \ell \le \lceil n/2 \rceil - 1$. Then the total number of $\CNOT$ gates to prepare $\ket{\psi}^{\otimes r}$ is upper bounded by

\begin{align*}
r \cdot \sum_{\ell = 1}^{\lceil n/2 \rceil - 1} 2^\ell + \sum_{\ell = \lceil n/2 \rceil}^{n-1} (1 + o(1)) 2^\ell
&\le r2^{\lceil n/2 \rceil} + (1 + o(1))2^n\\
&\le 2^{\lceil n/2 \rceil + o(n / \log n)} + (1 + o(1))2^n\\
&\le (1 + o(1))2^n\qedhere
\end{align*}

By <cit.>, an arbitrary multiplexor can be expressed as below.

\[
\Qcircuit @C=1em @R=.7em
& {/} \qw
& \controlu \qwx[1] \qw
& \qw
& \ccteq{2}
&& {/} \qw
& \controlu \qwx[1] \qw
& \controlu \qwx[1] \qw
& \controlu \qwx[1] \qw
& \qw
\\
& \qw
& \multigate{1}{\phantom{U}}
& \qw
& \ccteqg
&& \qw
& \controlu \qwx[1] \qw
& \gate{R_y}
& \controlu \qwx[1] \qw
& \qw
\\
& {/} \qw
& \ghost{U}
& \qw
& \ccteqg
&& {/} \qw
& \gate{\phantom{U}}
& \controlu \qwx[-1] \qw
& \gate{\phantom{U}}
& \qw
\\
\]

This decomposition is also valid when the multiplexor has $0$ select bits. A recursive application of this decomposition implies that an arbitrary $n$-qubit unitary may be expressed as a product of $2^n - 1$ different $(n-1,1)$-multiplexors, of which $2^{n-1} - 1$ are multiplexed $R_y$ gates, and the remaining $2^{n-1}$ are arbitrary multiplexors. Applying <Ref> and <Ref> to these multiplexors gives the desired bound.

§ CONCLUSION AND OUTLOOK

We have demonstrated that mass production phenomena are not unique to classical computation, and that they extend to quantum circuit complexity as well. As the message of this work is primarily conceptual in nature, we have not attempted to optimize every aspect of our results. Indeed, our mass production theorems could be extended further in a variety of ways; we outline a few such possibilities below.

If our results have any hope of being used in practice, then still more work needs to be done to optimize various constants. We suspect that the leading constant in <Ref> could be brought down from $5/2$ to $1$ with a more clever decomposition into multiplexors. The factors hidden in the $o(1)$ could probably be optimized further as well, especially those related to the constant factor $d$ that appears in <Ref>. Indeed, we believe that much of the redundancy in computing and uncomputing $\mathcal{B}_{n,k,\ell}$ for each $0 \le \ell \le 2^k$ could be reduced by more careful accounting.

It is also worth attempting to optimize other parameters of practical relevance, such as constraints on the gate set, locality, depth, and ancilla qubit count. In principle, our proof should allow for some tradeoff between depth and ancilla count, because the $\bar{g}_\ell$s in <Ref> can either be evaluated sequentially or in parallel. Another particularly interesting question is whether ancilla qubits are necessary at all to achieve quantum mass production.

We leave open the circuit complexity of quantum mass production in other parameter regimes. As <Ref> and <Ref> only apply when $r = 2^{o(n / \log n)}$, it is natural to ask what happens when $r$ is much larger. For Boolean functions, it is known that for any $n$-bit $f$, the “asymptotic complexity” of mass production $\lim_{r \to \infty} \frac{C(f^r)}{r}$ is bounded by $\poly(n)$ <cit.>, where $C(f^r)$ denotes the Boolean circuit complexity of implementing $r$ copies of $f$. However, it is unclear whether the same approach would generalize to quantum circuits.

Lastly, we ask: are there any restricted examples of quantum circuits that exhibit a mass production phenomenon? What about Clifford circuits? We observe if one allows implementation by non-Clifford gates, then $n$ copies of an arbitrary Clifford operation can be implemented by a circuit with at most $O(n^\omega)$ gates, where $\omega$ is the exponent of matrix multiplication. By the “canonical form theorem” of Aaronson and Gottesman <cit.>, every Clifford circuit can be expressed in the form H-C-P-C-P-C-H-P-C-P-C, where each letter corresponds to a layer of Hadamard, CNOT, or phase gates. The Hadamard and phase layers contain at most $O(n)$ gates total, so it suffices to show how to implement $n$ copies of a $\CNOT$ circuit using $O(n^\omega)$ gates. For any $M \in \mathbb{F}_2^{n \times n}$, define $U_M$ as the unitary transformation that acts as $U_M \ket{x}\ket{y} = \ket{x}\ket{y \oplus Mx}$ on computational basis states. As every $\CNOT$ circuit implements an invertible linear transformation $\ket{x} \rightarrow \ket{Mx}$ for some $M \in \mathbb{F}_2^{n \times n}$, a $\CNOT$ circuit can be implemented using $U_M$ and $U_{M^{-1}}$ and $O(n)$ additional gates via:

\[
\ket{x}\ket{0^n} \xrightarrow{U_M} \ket{x}\ket{Mx} \xrightarrow{U_{M^{-1}}} \ket{0^n}\ket{Mx} \xrightarrow{\mathrm{SWAP}} \ket{Mx}\ket{0^n}.
\]

Then, as in <Ref>, we can mass produce $U_M$ and $U_{M^{-1}}$ using fast matrix multiplication.

§ ACKNOWLEDGMENTS

Part of this work was done while the author attended the 2022 Extended Reunion for the Quantum Wave in Computing at the Simons Institute for the Theory of Computing. We thank Alex Meiburg for helpful discussions.


# The Lie algebra structure of the $HH^{1}$ of the blocks of the sporadic
Mathieu groups

William Murphy

###### Abstract.

Let $G$ be a sporadic Mathieu group and $k$ an algebraically closed field of
prime characteristic $p$, dividing the order of $G$. In this paper we describe
some of the Lie algebra structure of the first Hochschild cohomology groups of
the $p$-blocks of $kG$. In particular, letting $B$ denote a $p$-block of $kG$,
we calculate the dimension of $HH^{1}(B)$ and in the majority of cases we
determine whether $HH^{1}(B)$ is a solvable Lie algebra.

##  1\. Introduction

Let $G$ be a finite group and $k$ an algebraically closed field of prime
characteristic $p$ dividing the order of $G$. It is well known that the first
Hochschild cohomology group (the $HH^{1}$) of the group algebra $kG$ form a
Lie algebra over $k$. By a result of Fleischmann, Janiszczak and Lempken [11],
this Lie algebra has non-zero dimension over $k$. On the other hand, the
problem of determining whether $HH^{1}(B)\neq\\{0\\}$ for a $p$-block $B$ of
$kG$ is still open in general.

In this paper we settle this problem for the blocks of the sporadic Mathieu
groups, in particular showing the following.

######  Theorem 1.1.

Let $G$ be a sporadic Mathieu group. Then for all primes $p$ dividing $|G|$
and for all $p$-blocks $B$ of $kG$ with a non-trivial defect group,
$HH^{1}(B)\neq\\{0\\}$. Moreover, for each $B$ we determine the dimensions
$\mathrm{dim}_{k}(HH^{1}(B))$.

######  Remark 1.2.

In the process of proving Theorem 1.1 we collect interesting information on
the $p$-blocks of the sproadic Mathieu groups. Out of the collection of $40$
such $p$-blocks with non-zero defect, we determine that $37$ have $HH^{1}$
that is a solvable Lie algebra, and $1$ has $HH^{1}$ that is a simple Lie
algebra. Of the $18$ $p$-blocks whose $HH^{1}$ is a $2$-dimensional Lie
algebra, we determine that $15$ are non-abelian and $1$ is abelian. Moreover,
we have that for $G=M_{11}$ or $M_{22}$, $HH^{1}(kG)$ is a solvable Lie
algebra for all primes $p=\mathrm{char}(k)$ dividing $|G|$.

######  Remark 1.3.

By Farrell and Kessar [10, 8.1(ii)] the Morita-Frobenius number for the blocks
of the sporadic Mathieu groups is 1. Thus by [21] any basic algebra $A$ of
these blocks is defined over the prime field ${\mathbb{F}_{p}}$; $A$ has a
$k$-basis whose structure constants are contained in ${\mathbb{F}_{p}}$, or
equivalently for each block $B$ and basic algebra $A$ of $B$, there is an
${\mathbb{F}_{p}}$-algebra $A_{0}$ such that $A\cong
k\otimes_{{\mathbb{F}_{p}}}A_{0}$. Consequently, $HH^{1}(B)\cong
HH^{1}(A)\cong k\otimes_{{\mathbb{F}_{p}}}HH^{1}(A_{0})$ is an isomorphism of
Lie algebras; that is the Lie algebra $HH^{1}(B)$ is defined over
${\mathbb{F}_{p}}$.

Moreover, for $G=M_{11}$ or $M_{22}$ one can read of from Table 2 that the
blocks $B$ of $kG$ have a Frobenius number of 1, so that $B\cong
k\otimes_{k}B_{0}$ for some ${\mathbb{F}_{p}}$-algebra $B_{0}$: $B$ itself is
also defined over ${\mathbb{F}_{p}}$ and we see that in this case there is a
Lie algebra isomorphism $HH^{1}(B)\cong
k\otimes_{{\mathbb{F}_{p}}}HH^{1}(B_{0})$ .

In order to determine the dimensions of the $HH^{1}$ of the blocks of $kG$,
the dimension of $HH^{1}(kG)$ is calculated, and given in Table 1. In Section
2 we prove our result using the centraliser decomposition ([2, Theorem
2.11.2], reviewed in Proposition A.1) and the $p$-block structure of each
sporadic Mathieu group. In Section 3 we determine in general the Lie algebra
structure of $HH^{1}(B)$ for $B$ a block with a non-trivial cyclic defect
group, giving an explicit formula for its dimension (though this is already
folklore) and a characterisation of a basis for $HH^{1}(B)$ as a Lie algebra.
This is a solvable Lie algebra in general (see [30, Example 5.7]), and we note
that for some blocks $B$ with a non-trivial cyclic defect group, we see that
$HH^{1}(B)$ is $1$-dimensional. In this case the we are also able to determine
that there is $p$-toral basis of $HH^{1}(B)$.

In Section 4 we use an alternative method to determine the dimension of the
$HH^{1}$ of the non-principal $2$-block of the sporadic Mathieu group $M_{12}$
and we hope to use this method in other cases going forwards. When $G=M_{12}$
and $p=2$, the non-principal block of $kG$ has a non-cyclic and non-trivial
defect group; this is the only such case over all primes dividing the orders
of all sporadic Mathieu groups $G$, and so the only case where the centraliser
decomposition alone is insufficient to compute the dimensions of the $HH^{1}$.
In particular we use a result of Külshammer and Robinson [25] to verify this
dimension. This result gives an alternating sum formula for the dimension of
the Tate-Hochschild cohomology, equivalent to the Hochschild cohomology in
positive degree, and to our knowledge this paper marks the first explicit use
of this formula. Whilst a shorter proof exists to determine these dimensions
(see Theorem 2.4(i)), our verification showcases the power of this formula and
its potential application to other finite groups that do not have such “nice”
$p$-block structure as the Mathieu groups (see the remarks following Table 2).

Appendix A gives the GAP code constructed to compute the dimensions of the
$HH^{1}$ of (hypothetically) any finite group on which GAP can perform
calculations, using the well-known centraliser decomposition.

We note that although the Hochschild cohomology of a $k$-algebra $A$ is
defined by $HH^{*}(A)=\mathrm{Ext}^{*}_{A\otimes_{k}A^{\mathrm{op}}}(A;A)$,
the first Hochschild cohomology admits an alternative characterisation as the
set of $k$-linear derivations on $A$, modulo inner derivations:
$HH^{1}(A)\cong\mathrm{Der}(A)/\mathrm{IDer}(A)$. We will use this point of
view throughout.

We will write by $C_{n}$, $S_{n}$, $A_{n}$ and $D_{2n}$ the cyclic, symmetric,
alternating and dihedral groups of orders $n$, $n!$, $n!/2$ and $2n$
respectively, with $Q_{2^{n}}$ and $SD_{2^{n}}$ denoting the quaternion and
semi-dihedral groups of order $2^{n}$. The projective special linear group of
degree $n$ over the finite field with $q$ elements is denoted $L_{n}(q)$, and
for $G$ a group acting on a set $X$, we will denote by $X^{G}$ the fixed
points of $X$ under this action.

##  2\. The sporadic Mathieu groups

Let $G$ be a sporadic Mathieu group, and let $k$ be an algebraically closed
field of prime characteristic $p$ dividing $|G|$. Throughout we use GAP [12],
the ATLAS [5], the online group database found at [7], the Modular Atlas
project [38] and its print counterpart [18] as our sources of data on the
groups $G$, their subgroups and their blocks.

To calculate the dimensions of $HH^{1}(kG)$ over $k$ we use the centraliser
decomposition. From this we construct a simple code in GAP (see Appendix A)
which provides the following data on the dimensions of $HH^{1}(kG)$ at the
different primes $p$.

Table 1.  The dimensions of $HH^{1}(kG)$ for $p=\mathrm{char}(k)$ dividing $|G|$.  $G$ | Order | $p=2$ | $p=3$ | $p=5$ | $p=7$ | $p=11$ | $p=23$
---|---|---|---|---|---|---|---
$M_{11}$ | $2^{4}\cdot 3^{2}\cdot 5\cdot 11$ | $6$ | $2$ | $1$ | $-$ | $2$ | $-$
$M_{12}$ | $2^{6}\cdot 3^{3}\cdot 5\cdot 11$ | $15$ | $4$ | $2$ | $-$ | $2$ | $-$
$M_{22}$ | $2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11$ | $9$ | $3$ | $1$ | $2$ | $2$ | $-$
$M_{23}$ | $2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 23$ | $9$ | $5$ | $3$ | $4$ | $2$ | $2$
$M_{24}$ | $2^{10}\cdot 3^{3}\cdot 5\cdot 7\cdot 11\cdot 23$ | $22$ | $10$ | $4$ | $6$ | $1$ | $2$

We need to describe how the dimensions of $HH^{1}(kG)$ given in Table 1
decompose into the dimensions of the $HH^{1}$ of the $p$-blocks of $kG$, in
view of the structure provided in Table 2 below. This table - the data of
which one may verify in GAP \- describes the $p$-block structure of $kG$ for
the prime divisors $p$ of $|G|$, in particular the number of blocks and their
defect. The notation $[n_{0},n_{1},\ldots,n_{m}]$ denotes that $G$ has
$p$-block structure $kG=B_{0}\oplus B_{1}\oplus\cdots\oplus B_{m}$ where
$B_{0}$ is the principal block of $kG$, $B_{i}$ has defect $n_{i}$, and the
blocks are ordered according to the decreasing size of their defect. The
notation $1^{n}$ (resp. $0^{n}$) denotes $n$ successive $p$-blocks whose
defect groups are non-trivial cyclic (resp. trivial).

Table 2.  The defect of the $p$-blocks of $kG$ for $p=\mathrm{char}(k)$ dividing $|G|$.  $G$ | $p=2$ | $p=3$ | $p=5$ | $p=7$ | $p=11$ | $p=23$
---|---|---|---|---|---|---
$M_{11}$ | $[4,0^{2}]$ | $[2,0]$ | $[1,0^{5}]$ | $-$ | $[1,0^{3}]$ | $-$
$M_{12}$ | $[6,2]$ | $[3,1,0]$ | $[1^{2},0^{5}]$ | $-$ | $[1,0^{8}]$ | $-$
$M_{22}$ | $[7]$ | $[2,1,0^{3}]$ | $[1,0^{7}]$ | $[1,0^{7}]$ | $[1,0^{5}]$ | $-$
$M_{23}$ | $[7,0^{2}]$ | $[2,1,0^{5}]$ | $[1^{2},0^{8}]$ | $[1^{2},0^{7}]$ | $[1,0^{10}]$ | $[1,0^{4}]$
$M_{24}$ | $[10]$ | $[3,1^{4},0]$ | $[1^{3},0^{12}]$ | $[1^{3},0^{11}]$ | $[1,0^{15}]$ | $[1,0^{13}]$

The structure of the Mathieu groups lends itself to such calculations in
general: for each sporadic Mathieu group $G$ and each prime $p$ dividing
$|G|$, all non-principal $p$-blocks have a cyclic defect group aside from the
aforementioned exception of $G=M_{12}$ and $p=2$ (see the the last remark of
$\S 1$ in [1]). The blocks of the Mathieu groups with a non-trivial cyclic
defect group have structure that is well understood and the dimensions of the
$HH^{1}$ is easily calculated in this case (see Section 3). On the other hand
for the blocks with a trivial defect group the $HH^{1}$ vanishes: any such
block is isomorphic to a matrix algebra over $k$ (see [3]), and the $HH^{*}$
of a separable algebra is zero [13, Theorem 4.1].

The dimensions of the $HH^{1}$ of the principal block $B_{0}$ are then readily
calculated (aside from the noted exception) by comparing the dimensions of
$HH^{1}(kG)$ and $HH^{1}(B)$ for the non-principal blocks $B$ with cyclic
defect groups:

$\mathrm{dim}_{k}(HH^{1}(B_{0}))=\mathrm{dim}_{k}(HH^{1}(kG))\hskip
8.53581pt-\sum_{\begin{subarray}{c}B\neq B_{0},\\\ B\ \textnormal{has
cyclic}\\\ \textnormal{ defect
group}\end{subarray}}\mathrm{dim}_{k}(HH^{1}(B)).$

In the exceptional case, $G=M_{12}$ has a non-principal 2-block with Klein
four defect group and so to calculate the dimensions and show solvability of
the $HH^{1}$ of this block we use results of Holm [16], Erdmann [9] and
Linckelmann and Rubio y Degrassi [30].

Since we are concerned with determining not just the dimensions of the
$HH^{1}$ but its solvability as a Lie algebra, we will often make use of the
following result.

######  Theorem 2.1 ([30, 34]).

Let $A$ be a finite dimensional $k$-algebra such that
$\mathrm{dim}_{k}(\mathrm{Ext}^{1}_{A}(S,S))=0$ and
$\mathrm{dim}_{k}(\mathrm{Ext}^{1}_{A}(S,T))\leq 1$, for all non-isomorphic
simple $A$-modules $S$ and $T$. Then $HH^{1}(A)$ is a solvable Lie algebra.

We organise Theorem 1.1 into the following 5 results, categorised with regards
to which Mathieu group they relate to. In each theorem, $k$ is an
algebraically closed field of characteristic $p$ dividing $|G|$, $G$ a
sporadic Mathieu group. We will write $B_{0}$ throughout for the principal
$p$-block of $kG$, and $B_{i}$ for the non-principal $p$-blocks of $kG$,
$i=1,2,\ldots,$ if they exist, ordered in descending size of their defect. For
$P$ a cyclic $p$-group, and $E$ a cyclic $p^{\prime}$-group acting on $P$, we
let $HH^{1}(kP)^{E}$ denote the fixed points under the the action of $E$ on
$HH^{1}(kP)$, induced by the action of $E$ on $P$ (see Section 3 for details).
We adopt the convention that the “zero Lie algebra” ${\mathcal{L}}=\\{0\\}$
with Lie bracket $[0,0]=0$ is trivially solvable.

######  Theorem 2.2.

Let $G=M_{11}$. Then $HH^{1}(kG)$ is a solvable Lie algebra isomorphic to
$HH^{1}(B_{0})$, every non-principal block of $kG$ has defect 0, and the
following isomorphisms of Lie algebras with given dimensions hold.

  * (i)

Let $p=2$ and $B_{0}(kL_{3}(q))$ be the principal $2$-block of $L_{3}(q)$,
$q=3\ (\mathrm{mod}4)$. Then there are isomorphisms of $6$-dimensional
solvable Lie algebras,

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(B_{0}(kL_{3}(q))).$

  * (ii)

Let $p=3$. Then there are isomorphisms of $2$-dimensional Lie algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(k(C_{3}^{2}\rtimes SD_{16})).$

  * (iii)

Let $p=5$. Then there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{5})^{C_{4}}\cong k.$

  * (iv)

Let $p=11$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{11})^{C_{5}}.$

Table 3.  Data on the $p$-blocks of $kM_{11}$.  $G=M_{11}$ | $p=2$ | $p=3$ | $p=5$ | $p=11$
---|---|---|---|---
No. of blocks, $B_{i}$, of $kG$ | 3 | 2 | 6 | 4
Defects of the $B_{i}$ | $[4,0^{2}]$ | $[2,0]$ | $[1,0^{5}]$ | $[1,0^{3}]$
Dimensions $\mathrm{dim}_{k}(HH^{1}(B_{i}))$ | $[6,0^{2}]$ | $[2,0]$ | $[1,0^{5}]$ | $[2,0^{3}]$
Sylow $p$-subgroup $P$ of $G$ | $SD_{16}$ | $C_{3}^{2}$ | $C_{5}$ | $C_{11}$
$E=N_{G}(P)/PC_{G}(P)$ | $\\{1\\}$ | $SD_{16}$ | $C_{4}$ | $C_{5}$

In Table 3 above, and Tables 4, 5, 6, and 7 below, the notation of the second
row (the defects of the blocks) is the same notation as in Table 2. The
notation of the third row is similar: $[n_{0},n_{1},\ldots,n_{m}]$ denotes
that $G$ has $p$-block structure $kG=B_{0}\oplus B_{1}\oplus\cdots\oplus
B_{m}$ where $HH^{1}(B_{i})$ has dimension $n_{i}$, whilst $n^{m}$ denotes $m$
successive $p$-blocks whose $HH^{1}$ has dimension $n$.

###### Proof.

Let $G=M_{11}$. The proof will refer to facts included in Section 3.

By the data in Table 3 one sees that for all primes dividing $|G|$, the block
structure of $kG$ provides the isomorphism $HH^{1}(kG)\cong HH^{1}(B_{0})$.
The dimensions $\mathrm{dim}_{k}(HH^{1}(B_{0}))$ are therefore simply the same
as the dimensions of $HH^{1}(kG)$, which are calculated using the centraliser
decomposition.

Let $p=2$. A Sylow 2-subgroup of $G$ is isomorphic to the semidihedral group
of order 16, $SD_{2^{n}}=SD_{16}$, so that $B_{0}$ is a tame block. The
decomposition matrix of $B_{0}$ is available on GAP allowing confirmation that
$B_{0}$ has 3 irreducible Brauer characters, and so 3 simple modules up to
isomorphism. Using Erdmann’s classification of tame blocks [9, pp.300] one
verifies that the decomposition and Cartan matrices of $B_{0}$ match those of
$SD(3{\mathcal{B}})_{1}$. This may be further verified in [16, 3.3/3.4] which
gives us explicitly a derived equivalence between $B_{0}$ and
$B_{0}(L_{3}(q)),q=3\ (\mathrm{mod}4)$. The quiver and relations of $B_{0}$
given by Erdmann enable us to determine that $B_{0}$ is non-local, which
thanks to a result of Eisele and Raedschelders [8, 4.7] provides the
solvability of $HH^{1}(B_{0})$. This proves (i).

Now let $p=3$. A Sylow 3-subgroup $P$ is isomorphic to $C_{3}\times C_{3}$,
with inertial quotient $E=SD_{16}$ as in Table 3. By a result of Rouquier [33]
there is a stable equivalence of Morita type between $B_{0}$ and its Brauer
correspondent $kN_{G}(P)=P\rtimes E$, (in fact there is derived equivalence
between $B_{0}$ and $kN_{G}(P)$ constructed explicitly by Okuyama in an
unpublished result [32]). Whence $HH^{1}(B_{0})\cong HH^{1}(k(P\rtimes E))$
proving (ii). For (iii) and (iv), Theorems 3.1(iii) and 3.2 provide the second
isomorphism, and using Corollary 3.4 one may verify the dimensions of both
match those found by the centraliser decomposition, completing the proof. ∎

######  Remark 2.3.

In the sequel (Theorem 3.5) we will see that one can write down explicitly a
$k$-basis and Lie bracket for the Lie algebra $HH^{1}(kP)^{E}$ encountered in
Theorem 2.2(iii) and (iv), and that in fact this is possible for all
$p$-blocks of finite groups with a non-trivial cyclic defect group $P\cong
C_{p}$.

######  Theorem 2.4.

Let $G=M_{12}$. Then the following isomorphisms of Lie algebras with given
dimensions hold.

  1. (i)

Let $p=2$. Then $HH^{1}(B_{0})$ is a $13$-dimensional Lie algebra and there is
an isomorphism of $2$-dimensional abelian Lie algebras

$HH^{1}(B_{1})\cong HH^{1}(kA_{4}).$

  2. (ii)

Let $p=3$. Then $HH^{1}(B_{0})$ is a $3$-dimensional solvable Lie algebra and
there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(B_{1})\cong HH^{1}(kS_{3})\cong k.$

  3. (iii)

Let $p=5$. Then there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(B_{0})\cong HH^{1}(B_{1})\cong HH^{1}(C_{5})^{C_{4}}\cong k.$

  4. (iv)

Let $p=11$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{11})^{C_{5}}.$

######  Remark 2.5.

Let $p=3$. By Usami [36, Proposition 4] there is a perfect isometry between
the principal blocks of $kM_{12}$ and $L_{3}(3)$, and we note that the
$HH^{1}$ of both is a 3-dimensional solvable Lie algebra. The dimension of the
$HH^{1}$ in the $L_{3}(3)$ case is computed by the centraliser decomposition,
and to see solvability in this case we apply a result of Koshitani [23,
Theorem 1] to Theorem 2.1. On the other hand we are not aware of the existence
of a derived equivalence in the literature and so we are unable to determine
whether $HH^{1}(B_{0})$ and $HH^{1}(kL_{3}(3))$ are isomorphic as Lie
algebras.

Table 4.  Data on the $p$-blocks of $kM_{12}$ $G=M_{12}$ | $p=2$ | $p=3$ | $p=5$ | $p=11$
---|---|---|---|---
No. of blocks, $B_{i}$ of $kG$ | 2 | 3 | 7 | 9
Defects of the $B_{i}$ | $[6,2]$ | $[3,1,0]$ | $[1^{2},0^{5}]$ | $[1,0^{8}]$
Dimensions $\mathrm{dim}_{k}(HH^{1}(B_{i}))$ | $[13,2]$ | $[3,1,0]$ | $[1^{2},0^{5}]$ | $[2,0^{8}]$
Sylow $p$-subgroup $P$ of $G$ | $C_{4}^{2}\rtimes C_{2}^{2}$ | $3^{1+2}_{+}$ | $C_{5}$ | $C_{11}$
$E=N_{G}(P)/PC_{G}(P)$ | $\\{1\\}$ | $C_{2}\times C_{2}$ | $C_{4}$ | $C_{5}$

###### Proof.

As before, the proof refers to results from Section 3.

Let $p=2$. A defect group of $B_{1}$ is isomorphic to the Klein 4 group (see
[1]), so that $B_{1}$ is a tame block. The decomposition matrix of $B_{1}$ is
available on GAP allowing confirmation that $B_{1}$ has 3 irreducible Brauer
characters, and so 3 simple modules up to isomorphism. Using Erdmann’s
classification of tame blocks [9, pp.296] one verifies that the decomposition
and Cartan matrices of $B_{1}$ match those of $D(3{\mathcal{K}})$. We note
that by [30, Proposition 5.3] this implies that $HH^{1}(B_{1})$ is a solvable
Lie algebra. The isomorphism and dimension of $HH^{1}(B_{1})$ is then given by
Holm [16, 2.1(2)], and by Table 1 we recover
$\mathrm{dim}_{k}(HH^{1}(B_{0}))=13.$

To show that $HH^{1}(B_{1})$ is the abelian $2$-dimensional Lie algebra, we
use a technique that will occur again in the sequel (c.f. the proofs of
Theorems 2.6(ii) and 3.5) to find a basis for $HH^{1}(k(C_{2}\times
C_{2}))^{C_{3}}$, which we will see is isomorphic to $HH^{1}(kA_{4})$. Let
$P=C_{2}\times C_{2}$ have generators $x$ and $y$, and $E=C_{3}$ have a
generator $r$ so that $A_{4}\cong P\rtimes E$ with $r\cdot x=xy$ and $r\cdot
y=x$. One verifies that the action of $E$ on $P$ is Frobenius, so the
inclusion of $HH^{1}(kP)^{E}$ in $HH^{1}(k(P\rtimes E))$ (see Theorem 3.2
below) is an isomorphism. We then use a result of Le and Zhou [26, Corollory
3.4] to find a basis of size $8$ for $HH^{1}(kP)$, and a result of Linckelmann
[28, Proposition 2.5.10] to find a basis of size $2$ for $HH^{1}(kP)^{E}$ and
in particular we show that this basis has a trivial Lie bracket.

Let $P_{1}=\langle x\rangle$ and $P_{2}=\langle y\rangle$. Then the result
[26, Corollory 3.4] tells us that there is an isomorphism of Lie algebras
$HH^{1}(kP)\cong HH^{1}(kP_{1})\otimes_{k}kP_{2}\oplus
kP_{1}\otimes_{k}HH^{1}(kP_{2}).$ Let $d_{i}$ (resp. $d_{i}^{\prime}$) be the
derivation on $kP_{1}$ (resp. $kP_{2}$) sending $x\mapsto x^{i}$ (resp.
$y\mapsto y^{i}$) so that $\\{d_{0},d_{1}\\}$ is a $k$-basis for
$HH^{1}(kP_{1})$ as a Lie algebra (resp. $\\{d_{0}^{\prime},d_{1}^{\prime}\\}$
for $HH^{1}(kP_{2})$). A basis for $HH^{1}(kP)$ is then given by
$X=\\{d_{i}\otimes y^{j},x^{j}\otimes d_{i}^{\prime}\ |\ i,j=0,1\\}$, where
$d_{i}\otimes y^{j}$ sends $x\mapsto x^{i}y^{j}$ and $y\mapsto 0$, whilst
$x^{j}\otimes d_{i}^{\prime}$ sends $x\mapsto 0$ and $y\mapsto x^{j}y^{i}$.
Now let $f_{x}=d_{0}\otimes y$ and $f_{y}=x\otimes d_{0}^{\prime}$. Using the
result [28, Proposition 2.5.10] as inspiration, we then look for a basis of
$HH^{1}(kP)^{E}$ by considering the $E$-orbit sums of $X$ under the induced
action on $HH^{1}(kP)$, $\ {}^{z}D(g)=z\cdot D(z^{-1}\cdot g)$ for all $z\in
E,g\in P$ and $D\in HH^{1}(kP)$.

We denote by $\mathrm{Tr}_{1}^{E}:HH^{1}(kP)\to HH^{1}(kP)^{E}$ the relative
trace map, and define $D_{z}=\mathrm{Tr}_{1}^{E}(f_{z})$ for $z=x$ or $y$.
This gives the following $4$ maps; $D_{x}$ sends $x\mapsto 1+y$ and $y\mapsto
1+xy$, whilst $D_{y}$ sends $x\mapsto 1+xy$ and $y\mapsto 1+x,$ and one checks
that $\\{D_{x},D_{y}\\}$ forms a Lie algebra basis for $HH^{1}(kP)^{E}$. A
simple verification shows that $[D_{x},D_{y}]=0$, completing the proof of (i).

For the blocks in (iii) and (iv), and the non-principal block $B_{1}$ in (ii),
Theorems 3.1(iii) and 3.2 provide the isomorphisms, and Corollary 3.4 provides
the dimensions. The dimension of the $HH^{1}$ of the principal block in (ii)
is then verified using Table 1. To see that this principal block has $HH^{1}$
a solvable Lie algebra, we apply results of Koshitani and Waki [24, Theorem 1]
to Theorem 2.1: the former gives us the dimensions
$\mathrm{dim}_{k}(\mathrm{Ext}_{kG}^{1}(S,T))$ for all simple $kG$-modules $S$
and $T$, and this meets the solvability criteria of the latter.

∎

######  Theorem 2.6.

Let $G=M_{22}$. Then $HH^{1}(kG)$ is a solvable Lie algebra, and the following
isomorphisms of Lie algebras with given dimensions hold.

  1. (i)

Let $p=2$. Then $kG$ is indecomposable and $HH^{1}(kG)$ is a $9$-dimensional
solvable Lie algebra.

  2. (ii)

Let $p=3$. Then there is an isomorphism of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(B_{0})\cong HH^{1}(k(C_{3}^{2}\rtimes Q_{8})),$

and there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(B_{1})\cong HH^{1}(kS_{3})\cong k.$

  3. (iii)

Let $p=5$. Then there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(C_{5})^{C_{4}}\cong k.$

  4. (iv)

Let $p=7$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{7})^{C_{3}}.$

  5. (v)

Let $p=11$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{11})^{C_{5}}.$

Table 5.  Data on the $p$-blocks of $kM_{22}$.  $G=M_{22}$ | $p=2$ | $p=3$ | $p=5$ | $p=7$ | $p=11$
---|---|---|---|---|---
No. of blocks, $B_{i}$ of $kG$ | 1 | 5 | 8 | 8 | 6
Defects of the $B_{i}$ | $[7]$ | $[2,1,0^{3}]$ | $[1,0^{7}]$ | $[1,0^{7}]$ | $[1,0^{5}]$
Dimensions $\mathrm{dim}_{k}(HH^{1}(B_{i}))$ | $[9]$ | $[2,1,0^{3}]$ | $[1,0^{7}]$ | $[2,0^{7}]$ | $[2,0^{5}]$
Sylow $p$-subgroup $P$ of $G$ | $C_{2}^{4}\rtimes D_{8}$ | $C_{3}^{2}$ | $C_{5}$ | $C_{7}$ | $C_{11}$
$E=N_{G}(P)/PC_{G}(P)$ | $\\{1\\}$ | $Q_{8}$ | $C_{4}$ | $C_{3}$ | $C_{5}$

###### Proof.

Let $G=M_{22}$. As before the proof refers to Section 3.

By the data in Table 5 one sees that for all primes $p\neq 3$ dividing $|G|$
the block structure of $kG$ provides the isomorphism $HH^{1}(kG)\cong
HH^{1}(B_{0})$. The dimensions $\mathrm{dim}_{k}(HH^{1}(B_{0}))$ are therefore
simply the same as the dimensions of $HH^{1}(kG)$, which are calculated using
the centraliser decomposition.

In the case $p=2$, to see that $HH^{1}(kG)$ is a solvable Lie algebra, we use
the quiver for $B_{0}$ constructed by Hoffman [14] over ${\mathbb{F}}_{2}$
along with the fact that the simple $kG$-modules in $B_{0}$ have as a
splitting field ${\mathbb{F}}_{2}$ (see [17]) and the result follows by
Theorem 2.1. We note that in [27] Lempken and Staszewski construct the socle
series for the projective indecomposable modules in the principal block of the
3-fold cover of $M_{22}$, from which the result may also be deduced.

In the case $p=3$ we need only use Theorems 3.1 and 3.2, and Corollary 3.4 to
determine the isomorphisms $HH^{1}(B_{1})\cong HH^{1}(kS_{3})\cong
HH^{1}(kC_{3})^{C_{2}}$ and the Lie algebra dimension, which via Table 1
allows us to calculate
$\mathrm{dim}_{k}(HH^{1}(B_{0}))=\mathrm{dim}_{k}(HH^{1}(kG))-\mathrm{dim}_{k}(HH^{1}(B_{1}))=2$.
A Sylow 3-subgroup $P$ is isomorphic to $C_{3}\times C_{3}$, with inertial
quotient $E=Q_{8}$. As before, a result of Rouquier then [33] gives us a
stable equivalence of Morita type between $B_{0}$ and its Brauer correspondent
$kN_{G}(P)=P\rtimes E$, (and again there is derived equivalence between
$B_{0}$ and $kN_{G}(P)$ constructed explicitly by Okuyama in his unpublished
result [32]).

To show that $HH^{1}(B_{0})$ is the unique (up to isomorphism) non-abelian
2-dimensional Lie algebra, we generalise some ideas that will be seen in the
sequel, regarding blocks with a cyclic defect group, in particular Theorem
3.5. One verifies that the action of $E$ on $P$ is Frobenius, so the inclusion
of $HH^{1}(kP)^{E}$ in $HH^{1}(k(P\rtimes E))$ (Theorem 3.2 below) is an
equality. We use a result of Le and Zhou [26, Corollory 3.4] to find a basis
of size $18$ for $HH^{1}(kP)$, and a result of Linckelmann [28, Proposition
2.5.10] to find a basis of size $2$ for $HH^{1}(kP)^{E}$, and in particular we
show that this basis has a nontrivial Lie bracket.

Let $P$ have as generators $x$ and $y$, and $E$ be generated by $r$ and $s$
such that $sr=r^{3}s$, and write the action of $E$ on $P$ as $r\cdot x=y^{2}$,
$r\cdot y=x$, $s\cdot x=x^{2}y$ and $s\cdot y=xy$. Let $P_{1}=\langle
x\rangle$ and $P_{2}=\langle y\rangle$. Then the result [26, Corollory 3.4]
tells us that there is an isomorphism of Lie algebras $HH^{1}(kP)\cong
HH^{1}(kP_{1})\otimes_{k}kP_{2}\oplus kP_{1}\otimes_{k}HH^{1}(kP_{2}).$ Let
$d_{i}$ (resp. $d_{i}^{\prime}$) be the derivation on $kP_{1}$ (resp.
$kP_{2}$) sending $x\mapsto x^{i}$ (resp. $y\mapsto y^{i}$) so that
$\\{d_{0},d_{1},d_{2}\\}$ is a $k$-basis for $HH^{1}(kP_{1})$ as a Lie algebra
(resp. $\\{d_{0}^{\prime},d_{1}^{\prime},d_{2}^{\prime}\\}$ for
$HH^{1}(kP_{2})$). A basis for $HH^{1}(kP)$ is then given by
$X=\\{d_{i}\otimes y^{j},x^{j}\otimes d_{i}^{\prime}\ |\ i,j=0,1,2\\}$, where
$d_{i}\otimes y^{j}$ sends $x\mapsto x^{i}y^{j}$ and $y\mapsto 0$, whilst
$x^{j}\otimes d_{i}^{\prime}$ sends $x\mapsto 0$ and $y\mapsto x^{j}y^{i}$.
Now let $f_{x}=d_{0}\otimes y$ and $f_{y}=x\otimes d_{0}^{\prime}$. Using the
result [28, Proposition 2.5.10] as inspiration, we look for a basis of
$HH^{1}(kP)^{E}$ by considering the $E$-orbit sums of $X$ under the induced
action on $HH^{1}(kP)$, $\ {}^{z}D(g)=z\cdot D(z^{-1}\cdot g)$ for all $z\in
E,g\in P$ and $D\in HH^{1}(kP)$.

We denote by $\mathrm{Tr}_{1}^{E}:HH^{1}(kP)\to HH^{1}(kP)^{E}$ the relative
trace map, and define $D_{z}=\mathrm{Tr}_{1}^{E}(f_{z})$ for $z=x$ or $y$.
This gives the following. The map $D_{x}$ sends $x\mapsto
1+y+2x^{2}+2xy+xy^{2}+2x^{2}y^{2}$ and $y\mapsto
1+x^{2}+2y^{2}+xy+2x^{2}y+2xy^{2}$, whilst $D_{y}$ sends $x\mapsto
1+2x^{2}+y^{2}+xy+2xy^{2}+2x^{2}y$ and $y\mapsto
1+x+2y^{2}+2xy+x^{2}y+2x^{2}y^{2}$ and one checks that $\\{D_{x},D_{y}\\}$
does indeed form a Lie algebra basis for $HH^{1}(kP)^{E}$. A simple
verification shows that $[D_{x},D_{y}]=D_{y}-D_{x}$, completing the $p=3$
case.

For $p\in\\{5,7,11\\}$, Theorems 3.1(iii) and 3.2 provide the second
isomorphism and Corollary 3.4 provides the dimensions. ∎

######  Theorem 2.7.

Let $G=M_{23}$. Then the following isomorphisms of Lie algebras with given
dimensions hold.

  1. (i)

Let $p=2$. Then there is an isomorphism of solvable $9$-dimensional Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0}).$

  2. (ii)

Let $p=3$. Then there is an isomorphism of $2$-dimensional Lie algebras

$HH^{1}(B_{0})\cong HH^{1}(k(C_{3}^{2}\rtimes SD_{16})),$

and an isomorphism of $3$-dimensional simple Lie algebras

$HH^{1}(B_{1})\cong HH^{1}(kC_{3}).$

  3. (iii)

Let $p=5$. Then there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(B_{0})\cong HH^{1}(kC_{5})^{C_{4}},$

and there is an isomorphism of $2$-dimensional non-abelian Lie algebras

$HH^{1}(B_{1})\cong HH^{1}(kC_{5})^{C_{2}}.$

  4. (iv)

Let $p=7$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(B_{0})\cong HH^{1}(B_{1})\cong HH^{1}(kC_{7})^{C_{3}}.$

  5. (v)

Let $p=11$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{11})^{C_{5}}.$

  6. (vi)

Let $p=23$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{23})^{C_{11}}.$

Table 6.  Data on the $p$-blocks of $kM_{23}$ $G=M_{23}$ | $p=2$ | $p=3$ | $p=5$ | $p=7$ | $p=11$ | $p=23$
---|---|---|---|---|---|---
No. of blocks, $B_{i}$ of $kG$ | 3 | 7 | 10 | 9 | 11 | 14
Defects of the $B_{i}$ | $[7,0^{2}]$ | $[2,1,0^{5}]$ | $[1^{2},0^{8}]$ | $[1^{2},0^{7}]$ | $[1,0^{10}]$ | $[1,0^{4}]$
Dimensions $\mathrm{dim}_{k}(HH^{1}(B_{i}))$ | $[9,0^{2}]$ | $[2,3,0^{5}]$ | $[1,2,0^{8}]$ | $[2,2,0^{7}]$ | $[2,0^{10}]$ | $[2,0^{4}]$
Sylow $p$-subgroup $P$ of $G$ | $C_{2}^{4}\rtimes D_{8}$ | $C_{3}^{2}$ | $C_{5}$ | $C_{7}$ | $C_{11}$ | $C_{23}$
$E=N_{G}(P)/PC_{G}(P)$ | $\\{1\\}$ | $SD_{16}$ | $C_{4}$ | $C_{3}$ | $C_{5}$ | $C_{11}$

######  Remark 2.8.

Although the principal $2$-block of $kM_{22}$ and $kM_{23}$ have isomorphic
defect groups and $HH^{1}$ of equal dimension, they are not derived
equivalent: $kM_{22}$ and $kM_{23}$ have centres of dimension $12$ and $17$
respectively, whence the principal blocks have centres of dimension $12$ and
$15$ respectively.

###### Proof.

Let $G=M_{23}$. The proof refers to Section 3.

By the data in Table 6 one sees that for $p\in\\{2,11,23\\}$, the block
structure of $kG$ provides the isomorphism $HH^{1}(kG)\cong HH^{1}(B_{0})$ and
so the dimensions $\mathrm{dim}_{k}(HH^{1}(B_{0}))$ are therefore the same as
the dimensions of $HH^{1}(kG)$, calculated using the centraliser
decomposition. For the primes $p=11$ and $p=23$ this can be verified using the
isomorphisms of Theorems 3.1(iii) and 3.2, followed by Corollary 3.4. These
results also complete the case for $p=5$ or $7$.

To see that the principal $2$-block has $HH^{1}$ a solvable Lie algebra we use
[31, Theorem 4.7] which constructs the projective indecomposable
${\mathbb{F}}_{2}G$-modules and shows us that
$\mathrm{Ext}_{{\mathbb{F}}_{2}G}^{1}(S,S)=\\{0\\}$ and
$\mathrm{dim}_{{\mathbb{F}}_{2}}(\mathrm{Ext}_{{\mathbb{F}}_{2}G}^{1}(S,T))\leq
1$ for all simple ${\mathbb{F}}_{2}G$-modules $S$ and $T$. The field
${\mathbb{F}}_{2}$ is a splitting field for the simple $kG$-modules in $B_{0}$
(see [17]) and so by Theorem 2.1 we get that $HH^{1}(B_{0})$ is solvable.

Now let $p=3$. A Sylow 3-subgroup $P$ is isomorphic to $C_{3}\times C_{3}$,
with inertial quotient $E=SD_{16}$ as in Table 6. By a result of Rouquier [33]
there is a stable equivalence of Morita type between $B_{0}$ and its Brauer
correspondent $kN_{G}(P)=P\rtimes E$, (and as before there is derived
equivalence between $B_{0}$ and $kN_{G}(P)$ constructed explicitly by Okuyama
in an unpublished result). Whence $HH^{1}(B_{0})\cong HH^{1}(k(P\rtimes E))$,
the dimensions of which may be calculated using the centraliser decomposition.
The non-principal block $B_{1}$ with non-trivial defect has $1$ Brauer
character, so that the inertial quotient of $B_{1}$ is trivial, and by Theorem
3.1 the given isomorphism holds.

∎

######  Remark 2.9.

Let $p=3$. The isomorphism of $3$-dimensional simple Lie algebras
$HH^{1}(B_{1})\cong HH^{1}(kC_{3})$ agrees with results of Waki [37, Theorem
2.4] and Linckelmann and Rubio y Degrassi [30, Theorem 1.3]: the former shows
that $B_{1}$ contains a unique isomorphism class of simple $kG$-modules with
representative $S$, of dimension $231$, whose Loewy structure gives
$\mathrm{dim}_{k}(\mathrm{Ext}_{kG}^{1}(S,S))=1$. The latter then tells us
that if $HH^{1}(B_{1})$ is simple, then it must be nilpotent.

######  Theorem 2.10.

Let $G=M_{24}$. Then the following isomorphisms of Lie algebras with given
dimensions hold.

  1. (i)

Let $p=2$. Then $kG$ is indecomposable and $HH^{1}(kG)$ is a $22$-dimensional
solvable Lie algebra.

  2. (ii)

Let $p=3$. Then $\mathrm{dim}_{k}(HH^{1}(B_{0}))=6$ and there are isomorphisms
of $1$-dimensional Lie algebras

$HH^{1}(B_{i})\cong HH^{1}(kS_{3})\cong k$

for $i=1,\ldots,4$.

  3. (iii)

Let $p=5$. Then there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(B_{0})\cong HH^{1}(B_{1})\cong HH^{1}(kC_{5})^{C_{4}}\cong k,$

and there is an isomorphism of $2$-dimensional non-abelian Lie algebras

$HH^{1}(B_{2})\cong HH^{1}(kC_{5})^{C_{2}}.$

  4. (iv)

Let $p=7$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(B_{0})\cong HH^{1}(B_{1})\cong HH^{1}(B_{2})\cong
HH^{1}(kC_{7})^{C_{3}}.$

  5. (v)

Let $p=11$. Then there are isomorphisms of $1$-dimensional Lie algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong k.$

  6. (vi)

Let $p=23$. Then there are isomorphisms of $2$-dimensional non-abelian Lie
algebras

$HH^{1}(kG)\cong HH^{1}(B_{0})\cong HH^{1}(kC_{23})^{C_{11}}.$

######  Remark 2.11.

Let $p=3$. Similar to Remark 2.5, the result of Usami [36, Proposition 4]
gives a perfect isometry between the principal blocks of $kM_{24}$ and $He$,
the Held group. On the other hand no derived equivalence seems to exist in the
literature and so we cannot determine any further the Lie algebra structure of
$HH^{1}(B_{0}(kM_{24}))$

Table 7.  Data on the $p$-blocks of $kM_{24}$ $G=M_{24}$ | $p=2$ | $p=3$ | $p=5$ | $p=7$ | $p=11$ | $p=23$
---|---|---|---|---|---|---
No. of blocks, $B_{i}$, of $kG$ | 1 | 8 | 15 | 14 | 16 | 14
Defects of the $B_{i}$ | $[10]$ | $[3,1^{4},0]$ | $[1^{3},0^{12}]$ | $[1^{3},0^{11}]$ | $[1,0^{15}]$ | $[1,0^{13}]$
Dimensions $\mathrm{dim}_{k}(HH^{1}(B_{i}))$ | $[22]$ | $[6,1^{4},0]$ | $[1^{2},2,0^{12}]$ | $[2^{3},0^{11}]$ | $[1,0^{15}]$ | $[2,0^{13}]$
Sylow $p$-subgroup $P$ of $G$ | Order 1024 | $3_{+}^{1+2}$ | $C_{5}$ | $C_{7}$ | $C_{11}$ | $C_{23}$
$E=N_{G}(P)/PC_{G}(P)$ | $\\{1\\}$ | $D_{8}$ | $C_{4}$ | $C_{3}$ | $C_{10}$ | $C_{11}$

###### Proof.

Let $G=M_{24}$. For $p=2$ the case is clear, and we use the centraliser
decomposition to compute the dimension of $HH^{1}(B_{0})$. We note that from
GAP a Sylow $2$-subgroup of $G$ has a structural description isomorphic to
$(((D_{8}^{2}\rtimes C_{2})\rtimes C_{2})\rtimes C_{2})\rtimes C_{2}$.

To see that the principal $2$-block has $HH^{1}$ a solvable Lie algebra we use
[20, Section 5] which provides the socle series’ for the projective
indecomposable ${\mathbb{F}}_{2}G$-modules, from which it may be read that
$\mathrm{Ext}_{{\mathbb{F}}_{2}G}^{1}(S,S)=\\{0\\}$ and
$\mathrm{dim}_{{\mathbb{F}}_{2}}(\mathrm{Ext}_{{\mathbb{F}}_{2}G}^{1}(S,T))\leq
1$ for all simple ${\mathbb{F}}_{2}G$-modules $S$ and $T$. By James [17],
${\mathbb{F}}_{2}$ is a splitting field for $G$ and so by Theorem 2.1,
$HH^{1}(B_{0})$ is solvable.

Now let $p=3$. A Sylow 3-subgroup $P$ is isomorphic to $3_{+}^{1+2}$, the
extraspecial group of order 27 with exponent 3. The blocks $B_{i}$,
$i=1,\ldots,4$ have 2 Brauer characters so that the inertial quotients are
isomorphic to $C_{2}$ in this case, and the isomorphisms and given dimensions
hold by Theorems 3.1(iii) and 3.2, and Corollary 3.4. We calculate
$\mathrm{dim}_{k}(HH^{1}(B_{0}))=\mathrm{dim}_{k}(HH^{1}(kG))-\sum_{i=1}^{4}\mathrm{dim}_{k}(HH^{1}(B_{i}))$
using the centraliser and block decomposition of $kG$. This proves (ii).

For all remaining primes, we use the block structure and centraliser
decomposition to determine a counting argument, the number of Brauer
characters to determine the cyclic inertial quotient $E$, Theorems 3.1(iii)
and 3.2, and Corollary 3.4 to get the Lie algebra isomorphisms and dimensions
of the $HH^{1}$ of the blocks.

∎

###### Proof of Theorem 1.1.

The result follows from Theorems 2.2, 2.4, 2.6, 2.7 and 2.10. ∎

######  Remark 2.12.

If $G$ is a sporadic Mathieu group of order divisible by $p$ then by a result
of Fleischmann, Janiszczak and Lempken [11] $G$ has a $p$-element $x$ such
that $x\notin C_{G}(x)^{\prime}$, the commutator subgroup of $C_{G}(x)$. As
already stated this implies $HH^{1}(kG)\neq 0$ by the centraliser
decomposition, for if $x\notin C_{G}(x)^{\prime}$ then
$C_{G}(x)/C_{G}(x)^{\prime}$ has a direct factor that is a non-trivial
$p$-group so that $\mathrm{Hom}(C_{G}(x),k)\neq 0$ by Lemma A.3.

On the other hand, [11] also implies that for all 2-cocycles $\alpha\in
Z^{2}(G;k^{\times})$ the Hochschild cohomology group $HH^{1}(k_{\alpha}G)\neq
0$; we have a twisted group algebra analogue to the centraliser decomposition
(Proposition A.2) and for any group $H$, if $g\in H$ is a $p$-element then
$\alpha(h,g)=\alpha(g,h)$ for all $h\in C_{H}(g)$ (see [19, 2.6.1(iv)]) so
that the action of $C_{H}(g)$ on $V_{g,\alpha}$ in the twisted centraliser
decomposition of $HH^{1}(k_{\alpha}H)$ is trivial. In particular for each
sporadic Mathieu group $G$ there is a $p$-element $x\in G$ not contained in
$C_{G}(x)^{\prime}$ as determined in the result [11], and consequently there
is a direct summand equal to $H^{1}(C_{G}(x);V_{x,\alpha})$ in the twisted
centraliser decomposition of $HH^{1}(k_{\alpha}G)$ such that
$H^{1}(C_{G}(x);V_{x,\alpha})\cong\mathrm{Hom}(C_{G}(x),k)\neq 0$.

We note that more work is necessary to determine the block structure of the
$k_{\alpha}G$ and the dimensions of the $HH^{1}$ of $k_{\alpha}G$ and its
blocks.

##  3\. The $HH^{1}$ of blocks with nontrivial cyclic defect group

The theory of blocks with cyclic defect group is well understood (see for
example [6, 29, 35]) and the $HH^{1}$ of such blocks has a Lie algebra
structure that is particularly simple to describe.

Let $G$ be a finite group and $k$ an algebraically closed field of
characteristic $p$ dividing $|G|$. If $B=kGb$ is a block of $kG$ with block
idempotent $b$ and a defect group $P$, and if $C=kN_{G}(P)c$ is the Brauer
correspondent of $B$ for some idempotent $c$, then we form the inertial
quotient of $B$, the group $E=N_{G}(P,f)/PC_{G}(P)$ where $f$ is a block of
$kC_{G}(P)$ satisfying $fc=f$. The inertial quotient is unique up to
isomorphism, and so independent of choice of $f$. If $B=B_{0}$ is the
principal block, then $\mathrm{Br}_{P}(b)$ is the principal block of of
$kC_{G}(P)$, so that $f=c$ and $E=N_{G}(P)/PC_{G}(P)$. Note also that if $P$
is abelian, $PC_{G}(P)=C_{G}(P)$.

The inertial quotient appears frequently in the context of blocks with cyclic
defect groups, in view of the results stated below. Throughout, unless
otherwise stated $P$ will denote a nontrivial cyclic group of order $p$ and
$E$ a $p^{\prime}$-subgroup of $\mathrm{Aut}(P)$, of order $e$. Let $x$ and
$y$ be generators of $P$ and $E$ respectively, then $E$ acts on $P$ and we
will write the action as $y\cdot x=x^{s}$ for some $s=2,\ldots,p-1$, such that
$s^{e}=1(\mathrm{mod}\ p)$. Let $H=P\rtimes E$ denote the semidirect product.

The group algebra $kH$ is a Nakayama (also selfinjective serial) algebra (see
[29, §11.3]), and in particular, if $E$ is non-trivial, then by Linckelmann
and Rubio y Degrassi ([30, Example 5.7]) $HH^{1}(kH)$ is a solvable Lie
algebra with nilpotent derived Lie subalgebra.

######  Theorem 3.1.

[29, 11.1.1/3/11] Let $G$ be a finite group and $B$ be a block of $kG$ with a
nontrivial cyclic defect group $P$ and inertial quotient $E$, and let
$H=P\rtimes E$. Then the following hold.

  * (i)

The inertial quotient $E$ is cyclic of order dividing $p-1$, and acts freely
on $P$.

  * (ii)

There is an equality $|E|=|\mathrm{IBr}_{k}(B)|$.

  * (iii)

There is a derived equivalence between $B$ and $kH$.

Derived equivalences preserve Hochschild cohomology so that in particular,
with $B$ and $H$ as above, we have an isomorphism of restricted Lie algebras
$HH^{1}(B)\cong HH^{1}(kH)$. The group $H$ is a Frobenius group by virtue of
the free action of $E$ on $P$.

The next result is well-known and provides a characterisation of the $HH^{1}$
of blocks with a cyclic defect group in terms of the $HH^{1}$ of the group
algebra $k(P\rtimes E)$ and the derivations of $HH^{1}(kP)$ that are fixed
under the action of $E$ induced by the action on $P$. We point out that this
theorem holds more generally for $P$ an elementary abelian $p$-group, and
though the action of $E$ on $P$ is not necessarily Frobenius in this case the
result may still be exploited as was seen in the proofs of Theorems 2.4(i) and
2.6(ii).

######  Theorem 3.2.

Let $P$ be an elementary abelian $p$-group, $E\leq\mathrm{Aut}(P)$ and write
$G=P\rtimes E$. Then the action of $E$ on $P$ induces an action on
$HH^{1}(kP)$, and there is an inclusion of Lie algebras

$HH^{1}(kP)^{E}\hookrightarrow HH^{1}(kG).$

Moreover if the action of $E$ on $P$ is a Frobenius action then this inclusion
is an isomorphism.

###### Proof.

We show something slightly more general. Let $G$ be a finite group with $N$ a
normal subgroup of $p^{\prime}$-index, and recall that $HH^{*}(kG)\cong
H^{*}(G;kG)$ where $G$ acts on $kG$ by conjugation. By [4, III.10.4],
$H^{*}(G;kG)\cong H^{*}(N;kG)^{G/N}$, which by definition equals
$\mathrm{Ext}_{kN}^{*}(k,kG)^{G/N}$.

Since $N$ is of index coprime to $p$ we have a decomposition under the
conjugation action, $kG=kN\oplus k(G\backslash N)$, such that each summand is
stable under the action of $N$. One sees that

$\displaystyle HH^{*}(kG)$
$\displaystyle\cong\mathrm{Ext}_{kN}^{*}(k,kN)^{G/N}\oplus\mathrm{Ext}_{kN}^{*}(k,k(G\backslash
N))^{G/N}$
$\displaystyle=HH^{*}(kN)^{G/N}\oplus\mathrm{Ext}_{kN}^{*}(k,k(G\backslash
N))^{G/N}.$

Now if $N=P$ is an elementary abelian $p$-group and $G=P\rtimes E$, we have
$HH^{1}(kP)^{G/P}\cong HH^{1}(kP)^{E}\subseteq HH^{1}(kG)$.

We also have that if the action of $E$ on $P$ is Frobenius then $k(G\backslash
P)=\bigoplus_{y\in E\backslash\\{1\\}}kPy$ is a direct sum of projective
$kP$-modules under the conjugation action, so that
$\mathrm{Ext}_{kP}^{1}(k,k(G\backslash P))=0$, and the second statement
follows. ∎

Let $d\in HH^{1}(kP)$ be a derivation and recall that $x,y$ generate $P,E$
respectively, with $y\cdot x=x^{s}$ for some $s=2,\ldots,p-1$. Then the action
of $E$ on $HH^{1}(kP)$ in Theorem 3.2 is given by $\ {}^{y}d\in HH^{1}(kP)$,
the derivation sending $x\mapsto y\cdot d(y^{-1}\cdot x)$.

######  Proposition 3.3.

Let $P$ be an abelian $p$-group, $r$ the rank of $P/\Phi(P)$, and $E$ be an
abelian $p^{\prime}$-subgroup of $\mathrm{Aut}(P)$ acting freely on
$P\backslash\\{1\\}$. Set $H=P\rtimes E$. Then

$\mathrm{dim}_{k}(HH^{1}(kH))=r\cdot\left(\frac{|P|-1}{|E|}\right).$

###### Proof.

Since $E$ acts freely and is abelian, we have the $\frac{|P|-1}{|E|}+|E|$
conjugacy classes of $H$ as described in [35, §2], generalised to the
homocyclic case. Noting that $\mathrm{dim}_{k}(\mathrm{Hom}(P,k))=r$, the
result follows by the centraliser decomposition. ∎

From this we obtain the following simple corollary via the derived equivalence
of Theorem 3.1.

######  Corollary 3.4.

Let $B$ be a block of $kG$ with a nontrivial cyclic defect group $P$ and
nontrivial inertial quotient $E$. Then
$\mathrm{dim}_{k}(HH^{1}(B))=\frac{|P|-1}{|E|}$.

The following theorem provides us with a simple recipe to determine a Lie
algebra basis of the $HH^{1}$ of blocks with cyclic defect group.

######  Theorem 3.5.

Let $P$ be a cyclic group of order $q=p^{t}$ with a generator $x$, and $E$ a
$p$’-subgroup of $\mathrm{Aut}(P)$ of order $e$, with a generator sending $x$
to $x^{s}$. For $m=0,\ldots,q-1$, let $D_{m}$ denote the derivation on $kP$
that sends

$x\longmapsto\sum_{i=0}^{e-1}s^{i}x^{(m-1)s^{-i}+1}.$

Define an equivalence relation on $\\{D_{0},\ldots,D_{q-1}\\}$ by $D_{m}\sim
D_{n}$ if and only if $n=(m-1)s^{-i}+1$ for some $i=0,\ldots,e-1$. Then we
have the following.

  * (i)

If $E$ is non-trivial, then for all $m$, $D_{m}(x)\in J(kP)$.

  * (ii)

The set of equivalence class representatives of $\sim$ on
$\\{D_{0},\ldots,D_{q-1}\\}$ forms a $k$-basis for $HH^{1}(kP)^{E}$ as a Lie
algebra.

  * (iii)

Let $p>2$, $t=1$ so that $P$ is cyclic of order $p$, and let $E$ have order
$p-1$. Then $HH^{1}(kP)^{E}$ has a $p$-toral basis.

###### Proof.

The Automorphism group $\mathrm{Aut}(P)$ is isomorphic to the cyclic group
$C_{\varphi(q)}$ where $\varphi(q)=q-p^{t-1}$. Whence $E$ is cyclic acting
freely on $P$. Let $E$ have a generator $y$, so that $y\cdot x=x^{s}$. We have
the Lie algebra isomorphism $HH^{1}(kH)\cong HH^{1}(kP)^{E}$ of Theorem 3.2,
and we look for a basis of $HH^{1}(kP)^{E}$ of derivations $D\in HH^{1}(kP)$
such that ${}^{y^{j}}D(x^{i})=D(x^{i})$ for all $j=0,\ldots,e-1$ and
$i=0,\ldots,q-1$.

Let ${\mathcal{B}}=\\{d_{0},\ldots,d_{q-1}\\}$ be the basis of $HH^{1}(kP)$
given by the maps $d_{m}$ sending $x\mapsto x^{m}$. Taking inspiration from
[28, Proposition 2.5.10] we look for a basis of $HH^{1}(kP)^{E}$ in the set of
$E$-orbit sums of ${\mathcal{B}}$. We denote by
$\mathrm{Tr}_{1}^{E}:HH^{1}(kP)\to HH^{1}(kP)^{E}$ the relative trace map, and
define $D_{m}=\mathrm{Tr}_{1}^{E}(d_{m})$. This maps

$x\longmapsto\sum_{z\in E}z\cdot d_{m}(z^{-1}\cdot
x)=\sum_{i=0}^{e-1}y^{-i}\cdot d_{m}(x^{s^{i}}).$

One sees that each term in the sum may be expressed as
$y^{-i}\cdot(s^{i}x^{m+s^{i}-1})=s^{i}x^{(m-1)s^{-i}+1}$, as required. We note
that if $E$ is trivial then this formula recovers the basis ${\mathcal{B}}$ of
$HH^{1}(kP)$, for $D_{m}=d_{m}$ in this case.

Recall that $s$ is such that $s^{e}\equiv 1\ (\mathrm{mod}\ p)$. One sees that
$(1-s)(\sum_{i=0}^{e-1}s^{i})=1-s^{e}=0$, and if $E$ is non-trivial, then the
free action of $E$ on $P$ implies $s>1$, whence $\sum_{i=0}^{e-1}s^{i}=0$,
from which it is clear that $D_{1}\equiv 0$ and
$D_{m}(x)=\sum_{i=0}^{e-1}s^{i}x^{(m-1)s^{-i}+1}\in I(kP)=J(kP)$, proving (i).

One verifies that $\sim$ is indeed an equivalence relation on
$\\{D_{0},\ldots,D_{q-1}\\}$, and that there is a bijection between the set of
$E$-orbits on $P$ and the equivalence classes under $\sim$, via the map

$\displaystyle\\{E\cdot x^{j}\ |\ j=0,\ldots,q-1\\}$
$\displaystyle\leftrightarrow\\{[D_{m}]\ |\ m=0,\ldots,q-1\\},$
$\displaystyle\ E\cdot x^{m-1}$ $\displaystyle\mapsto[D_{m}],$

proving (ii).

Now suppose that $p\neq 2$, $t=1$ so that $P$ is cyclic of order $p$, and that
$E$ has order $p-1$. Fix an element $w=\sum_{i=1}^{p-1}i^{-1}x^{i}\in kP.$
Then $w\in I(kP)=J(kP)$, as $\sum_{i=1}^{p-1}i^{-1}=\sum_{i=1}^{p-1}i=0.$ On
the other hand $w\notin J(kP)^{2}$. Indeed $J(kP)^{2}$ is generated as
$k$-vector space by all elements of the form $x^{i}(x-1)^{2}$ as $i$ ranges
from $0$ to $p-1$, and if we set $w=\sum_{i=0}^{p-1}\lambda_{i}x^{i}(x-1)^{2}$
then on comparing coefficients we get a system of linear equations in the
variables $\lambda_{i}$ whose coefficient matrix has rank $p-2$ but whose
augmented matrix has rank $p-1$, whence no such $\lambda_{i}\in k$ exist. By
Nakayama’s Lemma $w$ generates $J(kP)$ as a $kP$-module, and noting that
$w^{p}=(\sum_{i=1}^{p-1}i^{-1}x^{i})^{p}=\sum_{i=1}^{p-1}i^{-p}x^{pi}=\sum_{i=1}^{p-1}i^{-1}=0$
one sees that $\\{1,w,w^{2},\ldots,w^{p-1}\\}$ is a basis for $kP$ as a
$k$-vector space.

One checks that $w$ is sent to $sw$ under the action of $y$ where $s$ is such
that $x$ is sent to $x^{s}$ by $y$ as before. By Proposition 3.4
$HH^{1}(kP)^{E}$ has dimension $1$ so we may choose $D_{0}$ as representative
of the class forming a basis, without loss of generality. Then we have that
$D_{0}(w)=w$. To see this, first note that $d_{0}(w)=\sum_{i=1}^{p-1}x^{i-1}$,
and consider

$D_{0}(w)=\sum_{z\in E}\ ^{z}d_{0}(w)=\sum_{i=0}^{p-2}y^{i}\cdot
d_{0}(y^{-i}\cdot
w)=\sum_{i=0}^{p-2}y^{i}\cdot\left(s^{-i}d_{0}(w)\right)=\sum_{i=0}^{p-2}\sum_{j=1}^{p-1}s^{-i}x^{(j-1)s^{i}},$

which on expanding the direct sum indexed by $j$ is seen to be equal to
$\sum_{i=1}^{p-2}i^{-1}w$. One verifies that $\sum_{i=1}^{p-2}i^{-1}=1$ and
the result then follows: $D^{p}_{0}$ sends $w$ to $w$.

∎

######  Remark 3.6.

In the previous result it is possible to show that for $P$ cyclic of order
$q=p^{t}$, $t>1$, there is still a basis of $HH^{1}(kP)^{E}$ consisting of
maps $D_{m}$ such that $D_{m}^{p}=D_{m}$. Indeed, fixing $w=\sum_{i=1,p\nmid
i}^{q-1}i^{-1}x^{i}$ one verifies by an analogous proof that
$\\{1,w,w^{2},\ldots,w^{q-1}\\}$ is a basis for $kP$ as a $k$-vector space,
and that $D_{m}^{p}(w)=D_{m}(w)=w$ for all $m=0,\ldots,q-1$, $m\neq 1$. On the
other hand a simple verification shows that in general $[D_{m},D_{n}]\neq 0$
for $n\neq m$ and so this basis is not $p$-toral.

##  4\. The Tate-Hochschild cohomology dimension formula

Using a result of Külshammer and Robinson [25, Theorem 1] we verify the
following statement from Theorem 2.4(i). As mentioned before we use this
alternative method for verification in order to demonstrate its use in
application to other groups and more general cases, and its power as a
computational tool.

######  Proposition 4.1.

Let $G=M_{12}$, let $k$ be an algebraically closed field of characteristic 2
and let $B_{0}$ be the principal block of $kG$. Then
$\mathrm{dim}_{k}(HH^{1}(B_{0}))=13$.

From Table 2 we see that $M_{12}$ is the only sporadic Mathieu group with a
non-principal block that has defect greater than 1, occuring when $p=2$. The
results of Section 3 no longer apply here and so the method used to find the
dimensions of the $HH^{1}$ of the principal block are different for this case
only. Fortunately we have the result of Holm ([16, 2.1]) to tell us that
$\mathrm{dim}_{k}(HH^{1}(B_{1}))=2$ in this case, from which we can use the
centraliser decomposition to calculate $\mathrm{dim}_{k}(HH^{1}(kG))=15$ and
recover $\mathrm{dim}_{k}(HH^{1}(B_{0}))=13$.

On the other hand, we can verify this dimension more explicitly, using a
formula for the dimension of the Tate-Hochschild cohomology given by
Külshammer and Robinson, developed from Knörr and Robinson’s refinement of
Alperin’s weight conjecture [22]. The results of this section demonstrate the
weight of application of this formula, and further applications along the same
vein have been used to verify the dimensions of the $HH^{1}$ of the principal
blocks given in this paper (we omit these details). Throughout let $G$ be
finite and $k$ be algebraically closed of characteristic $p$ dividing $|G|$.

In [22] Knörr and Robinson consider several simplicial complexes related to
the $p$-local structure of $G$, on which $G$ acts naturally by conjugation.
Let ${\mathcal{P}}_{G}$ denote the simplicial complex with $n$-simplices the
chains of non-trivial $p$-subgroups of $G$ of the form
$Q_{1}<Q_{2}<\cdots<Q_{n+1}$ (with strict inclusion). In the course of the
proof of Proposition 4.1, we will only consider
${\mathcal{U}}_{G}\subset{\mathcal{P}}_{G}$ of chains of the form
$Q_{1}<Q_{2}<\cdots<Q_{n}$ such that $Q_{i}=O_{p}(N_{G}(Q_{i}))$ for all
$i=1,\ldots,n$. We call such subgroups $Q_{i}$ radical p-subgroups.

We denote by $\sigma$ an element of ${\mathcal{P}}_{G}$ (also known as a
$p$-chain), that is $\sigma=Q_{0}<\cdots<Q_{m}$ for some integer $m\geq 0$,
such that $Q_{i}$ is a $p$-subgroup of $G$ for all $0\leq i\leq m$. The empty
chain is viewed as a $-1$-simplex, and so for this and also for notational
convenience all chains will be viewed as chains of the form
$Q_{0}<Q_{1}<\cdots<Q_{m}$ where it is always the case that
$Q_{0}=\\{1_{G}\\}$ (c.f [22, Section 2]).

Given a $p$-chain $\sigma$, we define the subgroups
$C_{G}(\sigma)=C_{G}(Q_{m})$ and $N_{G}(\sigma)=\cap_{i=0}^{m}N_{G}(Q_{i})$.
The length of $\sigma$ denoted $|\sigma|$, is equal to the number of
nontrivial subgroups in the chain: in our case it will always be equal to $m$
as our chains always start with the trivial subgroup. We say that $\sigma$ is
a radical or normal chain if $Q_{0}=O_{p}(G)$ and
$Q_{i}=O_{p}(N_{G}(\sigma_{i}))$, where $\sigma_{i}=Q_{0}<\cdots<Q_{i}$ for
all $0\leq i\leq m$. Denote by ${\mathcal{R}}_{G}$ the set of all radical
$p$-chains of $G$ (note this is not a simplicial complex, see the remarks
following [22, 3.4]).

Since $G$ acts by conjugation on ${\mathcal{P}}_{G}$ and all its subsets we
may consider ${\mathcal{P}}_{G}/G$, a set of $G$-conjugacy class
representatives of $p$-chains, with the analogous notation defined on the
subsets of ${\mathcal{P}}_{G}$ considered above.

For a $p$-subgroup $Q$ of $G$, denote by $\mathrm{Br}_{Q}:(kG)^{Q}\to
kC_{G}(Q)$ the Brauer homomorphism, sending $\sum_{g\in G}\lambda_{g}g$ to
$\sum_{g\in C_{G}(Q)}\lambda_{g}g$. With this notation, we have the following
result (restricted to positive degrees for Hochschild cohomology, c.f [29,
10.7.11]), in which we may replace ${\mathcal{P}}_{G}/G$ with either
${\mathcal{U}}_{G}/G$ or ${\mathcal{R}}_{G}/G$, without altering the result.

######  Theorem 4.2.

[25, Theorem 1] Let $G$ be a finite group and $b$ a block idempotent of $kG$.
For any chain $\sigma=Q_{0}<Q_{1}<\cdots<Q_{m}$ in ${\mathcal{P}}_{G}$ set
$B_{\sigma}=kN_{G}(\sigma)\mathrm{Br}_{Q_{m}}(b)$. Then for any integer $n>0$
we have

$\sum_{\sigma\in{\mathcal{P}}_{G}/G}(-1)^{|\sigma|}\mathrm{dim}_{k}(HH^{n}(B_{\sigma}))=0.$

In this alternating sum, the chain $\sigma=Q_{0}$ recovers the dimension
$\mathrm{dim}_{k}(HH^{1}(kGb))$ which we may isolate on one side of the
equality, and the other terms require only calculations of the dimensions of
the $HH^{1}$ of smaller group algebras, or blocks of smaller group algebras,
for which we can use the centraliser decomposition or Lemma 4.5 below.

Since Proposition 4.1 concerns the principal block of the $M_{12}$, we will
make use of the following when evaluating the Brauer homomorphism involved in
the structure of the $B_{\sigma}$ above.

######  Theorem 4.3.

[29, 6.3.14, Brauer’s Third Main Theorem] Let $G$ be a finite group and $b$
the principal block idempotent of $kG$. Then, for any $p$-subgroup $Q$ of $G$,
$\mathrm{Br}_{Q}(b)$ is the principal block of both $kC_{G}(Q)$ and
$kN_{G}(Q)$.

Similarly, the following result will often be used to show that the
$\mathrm{Br}_{Q_{m}}(b)$ in the structure of the $B_{\sigma}$ above is simply
equal to the identity of $kG$

######  Lemma 4.4.

[28, 1.11.4] Let $G$ be a $p$-group and $\mathrm{char}(k)=p$. Then $kG$ is
indecomposable.

For the remainder of this section we fix $G=M_{12}$, $\mathrm{char}(k)=2$, and
$B_{0}=kGb$ the principal 2-block of $kG$, with block idempotent $b$. In order
to prove Proposition 4.1 we will calculate some auxiliary $HH^{1}$ dimensions
that appear in the alternating sum. In particular, $G$ has a unique conjugacy
class of subgroups isomorphic to $C_{2}\times S_{5}$, to $S_{3}\times A_{4}$
and to $C_{2}^{2}\times S_{3}$, and the principal block of these groups arises
in the alternating sum as some $kN_{G}(\sigma)\mathrm{Br}_{Q}(b)$ for some
chain $\sigma$ and $2$-subgroup $Q$ of $G$.

######  Lemma 4.5.

Let $H$ be one of $C_{2}\times S_{5},\ S_{3}\times A_{4}$ or $C_{2}^{2}\times
S_{3}$. Then $kH$ has two blocks $B_{0}^{\prime}$ and $B_{1}^{\prime}$ and we
have the following dimensions.

  * (i)

Let $H=C_{2}\times S_{5}$ Then $\mathrm{dim}_{k}(HH^{1}(B_{0}^{\prime}))=22$
and $\mathrm{dim}_{k}(HH^{1}(B_{1}^{\prime}))=8.$

  * (ii)

Let $H=S_{3}\times A_{4}$. Then $\mathrm{dim}_{k}(HH^{1}(B_{0}^{\prime}))=12$
and $\mathrm{dim}_{k}(HH^{1}(B_{1}^{\prime}))=2.$

  * (iii)

Let $H=C_{2}^{2}\times S_{3}$. Then
$\mathrm{dim}_{k}(HH^{1}(B_{0}^{\prime}))=24$ and
$\mathrm{dim}_{k}(HH^{1}(B_{1}^{\prime}))=8.$

###### Proof.

First let $H=C_{2}\times S_{5}$. Blocks of $H$ are in bijection with blocks of
$kC_{2}\otimes_{k}kS_{5}$, and $kS_{5}$ decomposes into two blocks
$B_{0}(kS_{5})$ and $B_{1}(kS_{5})$ with defect groups isomorphic to $D_{8}$
and $C_{2}$ respectively. The block $B_{0}(kS_{5})$ is then a tame block of
type $D(2{\mathcal{A}})$, derived equivalent to $kS_{4}$ by Holm [15], which
by the centraliser decomposition has $\mathrm{dim}_{k}(HH^{1}(kS_{4}))=6$ and
has $\mathrm{dim}_{k}(HH^{0}(kS_{4}))=\mathrm{dim}_{k}(Z(kS_{4}))=5.$ One sees
therefore that $HH^{1}(B_{0}^{\prime})\cong HH^{1}(k(C_{2}\times S_{4}))$ and
the latter has an additive decomposition isomorphic to
$kC_{2}\otimes_{k}Z(kS_{4})\oplus kC_{2}\otimes_{k}HH^{1}(kS_{4})$ by the
Künneth formula (see [28, 2.21.7]), whence the dimension of
$HH^{1}(B_{0}^{\prime})$ is 22.

Similarly, the block $B_{1}(kS_{5})$ is derived equivalent to $kC_{2}$ by
Theorem 3.1(iii) so that $HH^{1}(kB_{0}^{\prime})\cong HH^{1}(k(C_{2}\times
C_{2}))$ which has dimension 8. Using the centraliser decomposition one
verifies that $HH^{1}(kH)$ has dimension 30.

Now let $H=S_{3}\times A_{4}$. The proof follows in the same vein as (i). The
group algebra $kA_{4}$ is indecomposable whereas $kS_{3}$ has 2 blocks
$B_{0}(kS_{3})$ and $B_{1}(kS_{3})$ with defect 1 and 0 respectively,
isomorphic to $kC_{2}$ and $M_{n}(k)$ respectively, for some $n$. Whence
$B_{0}^{\prime}\cong B_{0}(kS_{3}\otimes_{k}kA_{4})\cong k(C_{2}\times A_{4})$
which has $HH^{1}$ of dimension 12 by the centraliser decomposition. Similarly
$B_{1}^{\prime}\cong kA_{4}$ which has $HH^{1}$ of dimension 2, and using the
centraliser decomposition one verifies that $HH^{1}(kH)$ has dimension 14.

Finally let $H=C_{2}^{2}\times S_{3}$. One sees that $B_{0}^{\prime}\cong
B_{0}(kC_{2}^{2}\otimes_{k}kS_{3})\cong
kC_{2}^{2}\otimes_{k}B_{0}(kS_{3})\cong kC_{2}^{3}$ which has $HH^{1}$ of
dimension 24, $B_{1}^{\prime}\cong kC_{2}^{2}$ which has $HH^{1}$ of dimension
8, and using the centraliser decomposition one verifies that $HH^{1}(kH)$ has
dimension 32. ∎

###### Proof of Proposition 4.1.

We will apply Külshammer and Robinson’s result using ${\mathcal{U}}_{G}$, the
complex of radical 2-subgroups of $G$. A defect group of $B_{0}$ is given by
$P\cong C_{4}^{2}\rtimes C_{2}^{2}$ (see SmallGroup(64,134) in the GAP
library), and using [1, Table 2D] we deduce the information displayed in Table
8, which gives us the radical 2-subgroups of $G$, up to $G$-conjugacy (the
details of the actions of the semidirect product may be found using GAP and
the database [7]).

Table 8. The radical 2-subgroups of $M_{12}$ Label, $Q$ |  Isomorphic to |  $N_{G}(Q)$ |  $C_{G}(Q)$
---|---|---|---
$P_{0}$ |  $\\{1_{G}\\}$ |  $G$ |  $G$
$P_{1}$ |  $C_{2}$ |  $C_{2}\times S_{5}$ |  $C_{2}\times S_{5}$
$P_{2}$ |  $C_{2}\times C_{2}$ |  $S_{3}\times A_{4}$ |  $C_{2}^{2}\times S_{3}$
$P_{3}$ |  $C_{4}\rtimes D_{8}$ |  $C_{4}^{2}\rtimes D_{12}$ |  $C_{2}\times C_{2}$
$P_{4}$ |  $2_{+}^{1+4}$ |  $Q_{8}\rtimes S_{4}$ |  $C_{2}$
$P$ |  $C_{4}^{2}\rtimes C_{2}^{2}$ |  $P$ |  $C_{2}$

The $G$-conjugacy classes of subgroups of $P$ and their lattice are given in
Figure 1, where reading left to right a connecting line denotes strict
inclusion.

Figure 1. Lattice of radical 2-subgroups of $M_{12}$
${{P_{0}}}$${{P_{1}}}$${{P_{3}}}$${P}$${{P_{2}}}$${{P_{4}}}$

We therefore have that ${\mathcal{U}}_{G}/G$ consists of 20 chains of radical
2-subgroups (all starting at $P_{0}=\\{1_{G}\\}$); one of length 0, five of
length 1, eight of length 2, 5 of length 3 and one of length 4. From these 20
chains, we obtain the information displayed in Table 9. We note that the
$B_{\chi}$ are found using Theorem 4.3, Lemma 4.4 and GAP, the dimensions
$\mathrm{dim}_{k}(HH^{1}(B_{\chi}))$ for $\chi\in\\{\beta,\gamma,\eta\\}$ are
calculated in Lemma 4.5, and all other dimensions are calculated using the
centraliser decomposition.

Table 9. Representatives of the classes of ${\mathcal{U}}_{G}/G$ $\chi$ |  Representative |  $B_{\chi}$ |  $\mathrm{dim}_{k}(HH^{1}(B_{\chi}))$
---|---|---|---
$\alpha$ |  $P_{0}$ |  $B_{0}(kG)=kGb$ |  ?
$\beta$ |  $P_{0}<P_{1}$ |  $B_{0}(k(C_{2}\times S_{5}))$ |  22
$\gamma$ |  $P_{0}<P_{2}$ |  $B_{0}(k(S_{3}\times A_{4}))$ |  12
$\delta$ |  $P_{0}<P_{3}$ |  $k(C_{4}^{2}\rtimes D_{12})$ |  30
$\epsilon$ |  $P_{0}<P_{4}$ |  $k(Q_{8}\rtimes S_{4})$ |  23
$\zeta$ |  $P_{0}<P$ |  $kP$ |  40
$\eta$ |  $P_{0}<P_{1}<P_{2}$ |  $B_{0}(k(C_{2}^{2}\times S_{3}))$ |  24
$\theta$ |  $P_{0}<P_{1}<P_{4}$ |  $k(C_{2}\times D_{8})$ |  28
$\iota$ |  $P_{0}<P_{1}<P$ |  $k(C_{2}\times D_{8})$ |  28
$\kappa$ |  $P_{0}<P_{1}<P_{3}$ |  $k(C_{2}\times S_{4})$ |  22
$\lambda$ |  $P_{0}<P_{2}<P_{4}$ |  $k(C_{2}\times A_{4})$ |  12
$\mu$ |  $P_{0}<P_{2}<P$ |  $kC_{2}^{3}$ |  24
$\nu$ |  $P_{0}<P_{4}<P$ |  $kP$ |  40
$\xi$ |  $P_{0}<P_{3}<P$ |  $kP$ |  40
$\pi$ |  $P_{0}<P_{1}<P_{2}<P_{4}$ |  $kC_{2}^{3}$ |  24
$\rho$ |  $P_{0}<P_{1}<P_{2}<P$ |  $kC_{2}^{3}$ |  24
$\sigma$ |  $P_{0}<P_{1}<P_{4}<P$ |  $k(C_{2}\times D_{8})$ |  28
$\tau$ |  $P_{0}<P_{1}<P_{3}<P$ |  $k(C_{2}\times D_{8})$ |  28
$\phi$ |  $P_{0}<P_{2}<P_{4}<P$ |  $kC_{2}^{3}$ |  24
$\psi$ |  $P_{0}<P_{1}<P_{2}<P_{4}<P$ |  $kC_{2}^{3}$ |  24

Let $\Lambda=\\{\beta,\gamma,\ldots,\psi\\}$, that is, the set of all chain
labels $\chi$ in Table 9, excluding $\chi=\alpha$. Then the alternating sum
formula of Theorem 4.2 tells us that

$\mathrm{dim}_{k}(HH^{1}(B_{\alpha}))=\mathrm{dim}_{k}(HH^{1}(B_{0}(kG)))=\sum_{\chi\in\Lambda}(-1)^{|\chi|+1}\mathrm{dim}_{k}(HH^{1}(B_{\chi}))=13.$

This completes the proof.

∎

##  Appendix A The dimension of the $HH^{1}$ of a finite group algebra

The much celebrated centraliser decomposition allows us to determine the
dimensions of the $HH^{1}$ of a finite group algebra with relative ease. It’s
heavy use throughout the calculations and proofs in this paper confirms its
utility, and so we explain briefly where these dimensions come from, as well
as detail the GAP code used to efficiently implement the centraliser
decomposition and compute these dimensions.

######  Proposition A.1.

[2, Theorem 2.11.2(ii)] As a $k$-vector space, the additive structure of the
Hochschild cohomology of $kG$ is given by

$HH^{*}(kG)\cong\bigoplus_{x\in G/\sim}H^{*}(C_{G}(x);k)$

where $G/\sim$ is the set of conjugacy class representatives of $G$ and
$C_{G}(x)$ is the centraliser of $x\in G$.

The first cohomology group of $G$ with coefficients in $k^{\times}$,
$H^{1}(G;k^{\times})$, will always be considered with trivial action of $G$ on
$k^{\times}$. One therefore sees that

$HH^{1}(kG)\cong\bigoplus_{x\in G/\sim}\mathrm{Hom}(C_{G}(x),k)$

where each summand is a finite abelian group of order dividing $|C_{G}(x)|$,
and in particular when viewed as $k$-modules,
$\mathrm{dim}_{k}(HH^{1}(kG))=\sum_{x\in
G/\sim}\mathrm{dim}_{k}(\mathrm{Hom}(C_{G}(x),k))$.

There exists a twisted group algebra analogue to the centraliser decomposition

######  Proposition A.2.

[39, Lemma 3.5] There is an additive decomposition

$HH^{1}(k_{\alpha}G)\cong\bigoplus_{x\in G/\sim}H^{1}(C_{G}(x);V_{x,\alpha}),$

where $V_{x,\alpha}$ is a $1$-dimensional $C_{G}(x)$-module spanned by the
image $\hat{x}\in k_{\alpha}G$ of $x\in G$, with action given by
$g\cdot\hat{x}=\alpha(g,x)\alpha(x,g)^{-1}\hat{x}$ for all $g\in C_{G}(x)$.

To calculate the individual dimensions of $\mathrm{Hom}(C_{G}(x),k)$ in
Proposition A.1 we use the next result, which then forms the rest of the GAP
code.

######  Lemma A.3.

The $k$-vector space $\mathrm{Hom}(G,k)$ is nonzero if and only if $G$ has a
quotient isomorphic to a nontrivial $p$-group. In particular,
$\mathrm{dim}_{k}(\mathrm{Hom}(G,k))=\mathrm{dim}_{k}(\mathrm{Hom}(R/\Phi(R),k))$,
where $R=G/O^{p}(G)$.

We detail below the simple GAP commands to determine the dimensions of the
$HH^{1}$ of a finite group algebra (in particular a group on which GAP is able
to perform calculations), using the centraliser decomposition and Lemma A.3.

Once a group $G$ is defined in GAP, the command div below gives the set of
prime divisors of $|G|$, $\pi(G)$. The functions following div then take as
input either $G$, or $G$ and a prime $p\in\pi(G)$, and output the following
lists;

  * 1.

The centralisers of a complete set of conjugacy class representatives of $G$,

  * 2.

For each conjugacy class representative $x\in G/\sim$, the groups
$C_{G}(x)/O^{p}(C_{G}(x))=R$,

  * 3.

The elementary abelian $p$-groups $R/\Phi(R)$,

  * 4.

The rank of $R/\Phi(R)$,

respectively. The final function then produces a list of dimensions
$\mathrm{dim}_{k}(HH^{1}(kG))$, with one entry for each $p\in\pi(G)$.

`div:=PrimeDivisors(Size(G));`

`ListFpCentralisers:=function(X);`

`> return List(ConjugacyClasses(X),i->`

`Image(IsomorphismFpGroup(Centralizer(X,Representative(i)))));`

`> end;;`

`ListMaxPQuot:=function(X,p);`

`> return List(ListFpCentralisers(X),i -> Image(EpimorphismPGroup(i,p)));`

`> end;;`

`ListElAb:=function(X,p);`

`> return List(ListMaxPQuot(X,p), i -> i/FrattiniSubgroup(i));`

`> end;;`

`ListPRank:=function(X,p);`

`> return List(ListElAb(X,p), i -> RankPGroup(i));`

`> end;;`

`dimHH1:=function(X);`

`> return List(div,i->Sum(ListPRank(X,i)));`

`> end;`

Acknowledgements. I would like to thank Xin Huang for his helpful comments on
an earlier version of this paper, and my PhD supervisor Markus Linckelmann for
his support, and for many interesting conversations.

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