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\title{Lecture 4-10: Clifford algebras}
\date{April 10, 2024}
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We digress to study the representations of a particular family of groups, which I privately call the Clifford groups, in detail.   This will lead to an important class of algebras universally called (complex) Clifford algebras, which are of considerable independent interest and importance.   We will also study real Clifford algebras, which exhibit an even richer structure.
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We begin with the group $G_n$.  It is generated by $n$ elements $a_1,\ldots,a_n$, with the defining relations $a_i^2=\epsilon,\epsilon^2=1$ and $a_ia_j=\epsilon a_ja_i$ for $i\ne j$.   Note that if $n=2$ this is just the group of quaternion units, generated by the quaternions $i$ and $j$, with $\epsilon=-1$.  In general there does not seem to be any standard name for this group in the literature; the term \lq\lq Clifford" group is probably as good as any other.  The center $Z_n$ of $G_n$ has order 2 and is generated by $\epsilon$ if $n$ is even; it is generated by $\epsilon$ and $z_n=a_1\ldots a_n$ if $n$ is odd.  More precisely, $Z_n$ is cyclic of order 4 if $n\equiv1\mod 4$; it is isomorphic to the Klein four-group if $n\equiv3\mod 4$.  For any $g\in G_n$, the conjugacy class of $g$ consists of $g$ alone if $g\in Z_n$ and of $g$ and $g\epsilon$ if $g\notin Z_n$.  Thus $G_n$ has $2^n+1$ conjugacy classes if $n$ is even and $2^n+2$ conjugacy classes if $n$ is odd.
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We first dispose of the boring representations of $G_n$; that is, those of dimension 1.  These are the ones for which $\epsilon$ acts trivially.  The quotient $G_n/\langle\epsilon\rangle$ of $G_n$ by the subgroup generated by $\epsilon$ is the direct product of $n$ copies of the cyclic group $\mathbb Z_2$; accordingly $G$ has $2^n$ one-dimensional representations.
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There is room for only one more irreducible representation of $G_n$ if $n$ is even, necessarily of degree $2^{\frac n 2}$, since the sum of the squares of the irreducible degrees is then $2^n+2^n=2^{n+1}$. the order of $G_n$.  If $n$ is odd, there are two more irreducible representations; since each has degree a power of 2 and the sum of the squares of the degrees must again be $2^{n+1}$, both must have degree $2^{\frac {n-1} 2}$.  In all of these cases $\epsilon\in G_n$ acts by a scalar square root of 1, so it must act by $-1$.
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The complex \alert{Clifford algebra} $C_n$ is then defined to be the quotient of the group algebra $\mathbb CG_n$ by the ideal generated by the central element $\epsilon+1$.  This is a single matrix algebra $M_{\frac n 2}(\mathbb C)$ if $n$ is even and the direct sum $M_{\frac {n-1} 2}(\mathbb C)\oplus M_{\frac {n-1} 2}(\mathbb C)$ if $n$ is odd.  It can also be thought of the algebra over $\mathbb C$ generated by $a_1,\ldots,a_n$ with the defining relations $a_i^2=-1,a_ia_j=-a_ja_i$ for $i\ne j$.   Sometimes one takes $a_i^2$ to be 1 rather than $-1$; this leads to an isomorphic algebra.
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If $n$ is odd, there is a very simple relationship between the two irreducible representations of degree $2^{\frac {n-1} 2}$.  If $n\equiv3\mod 4$, then $G_n$ is isomorphic to the direct product of $G_{n-1}$ and the cyclic subgroup $\langle z_n\rangle$ generated by $z_n$, which has order 2.  Starting from the unique irreducible representation $V_{n-1}$ of $G_{n-1}$ of degree larger than one, we extend it to $G_n$ by having $z_n$ act either trivially or by $-1$, thus obtaining the two representations $V_n,V_n'$ of $G_n$ of degree larger than one.  If instead $n\equiv1\mod 4$, then $z_n^2=\epsilon$, so $z_n$ has order 4.  Starting with $V_{n-1}$ as before, extend it to $G_n$ by having $z_n$ act by either $\pm\sqrt{-1}\in\mathbb C$, once again obtaining $V_n$ and $V_n'$.
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 If $n$ is even, then the representation $V_n$ turns out to decompose over $G_{n-1}$ as the sum of the two representations $V_{n-1},V_{n-1}'$.  We will see later that any representation of a subgroup $H$ of a group $G$ can be \lq\lq induced" to a larger representation of $G$; inducing either $V_{n-1}$ or $V_{n-1}'$ from $G_{n-1}$ to $G_n$, one realizes the representation $V_n$.  The character table of $G_n$ is easy to compute, since any non-central $g\in G_n$ is conjugate to $\epsilon g$, which acts by the negative of the action of $g$ on any irreducible representation of degree larger than one, whence its character on that representation is 0.
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We get even more interesting behavior by replacing the basefield $\mathbb C$ with $\mathbb R$.  In general, the {\sl real} group algebra $\mathbb RG$ of a finite group $G$ is not a direct sum of matrix rings over $\mathbb R$; instead it is a sum of matrix rings over (any or all) of $\mathbb R,\mathbb C$, and the quaternions $\mathbb H$.  For example, defining the real Clifford algebra $R_n$ by the same generators and relations as above, we see that $R_2$ satisfies the same relations as $\mathbb H$, so is isomorphic to it.  In general $R_n$ is either a single matrix algebra or the sum of two isomorphic ones.  Its structure (that is, whether there are one or two matrix algebras and whether they are real, complex, or quaternionic) is periodic with period eight; all of this is related to a topological phenomenon called {\sl Bott periodicity}.
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The real Clifford algebra $C_n$ is not a group under multiplication, but it has a large subset which is a group, isomorphic to the simply connected double cover Spin$(n,\mathbb R)$ of the special orthogonal group $SO(n,\mathbb R)$.  Similarly, the complex Clifford algebra $C_n$ has a copy of the double cover Spin$(n,\mathbb C)$ of $SO(n,\mathbb C)$ sitting inside it.
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Finally, I mention the {\sl composition problem}, which asks for which positive integers $n$ there exist real numbers $b_{ijk}$ for $1\le i,j,k\le n$ such that for all $x_1,\ldots,x_n,y_1,\ldots y_n\in\mathbb R$ such that if we set $z_i = \sum_{j,k}b_{ijk}xjy_k$ then we have an identity $(\sum_ix_i^2)(\sum_i y_i^2) = \sum_i z_i^2$.  It turns out that such an identity exists for exactly four values of $n$, namely 1,2,4, and 8.  The identity for $n=1$ is trivial.  For $n=2$ it comes to the multiplicativity of the complex norm; likewise for $n=4$ it comes down to the multiplicativity of a norm, this time the quaternionic norm.  For $n=8$ there is another set of numbers called the {\sl octonions} admitting a multiplicative real-valued norm.  The octonions are even worse behaved than the quaternions:  in addition to being noncommutative they are not even associative under multiplication.  This roughly explains why one does not just iterate the procedure to get the quaternions from the complex numbers ad infinitum.
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That the composition problem has no solution for any $n$ other than $1,2,4$, or $8$ comes down to a calculation using the representation theory of $R_n$.
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