Schur Indices and Fields of Definition

Let $G$ be finite and let $\chi$ be the character of a complex representation of $G$. Its character field is $\mathbb{Q}(\chi ):=\mathbb{Q}(\{\chi (g):g\in G\})\subseteq \mathbb{C}\mathrm{.}$ More generally, if $F\subseteq \mathbb{C}$, write $F(\chi )$ for the field generated by $F$ and the values of $\chi$ in the sense of Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions.

If $F\subseteq \mathbb{C}$, a complex representation $\rho :G\to \mathrm{GL}(V)$ is realizable over $F$, and $F$ is a field of definition for $\rho$, when there is a finite-dimensional representation ${\rho }_F:G\to \mathrm{GL}(W)$ over $F$ such that $\mathbb{C}{\otimes }_{F}W$ is equivalent to $V$. Thus a character field records traces, whereas a field of definition records a matrix model; neither term asserts that the other field has the other property.

Let $E\mathrm{/}F$ be finite Galois, let $G$ be a group, and let $V$ be an $E$-representation with matrices $\rho (g)$ in an $E$-basis. For $\sigma \in \mathrm{Gal}(E\mathrm{/}F)$, the $\sigma$-conjugate ${}^{\sigma }V$ is the $E$-representation whose matrix for $g$ is obtained by applying $\sigma$ entrywise to $\rho (g)$. This does not depend on the chosen basis up to equivalence: conjugating every entry of $A\rho (g)A^{-1}$ gives $\sigma (A)\sigma (\rho (g))\sigma (A{)}^{-1}$.

Its character is ${\chi }_{{}^{\sigma }V}=\sigma \circ {\chi }_V$. We write $\mathrm{Stab}(V)$ for the subgroup of Galois automorphisms for which ${}^{\sigma }V\cong V$.

Let $E\mathrm{/}F$ be a field extension, $G$ a group, and $V,W$ finite-dimensional $F$-representations of $G$. The map $E{\otimes }_F{\mathrm{Hom}}_G(V,W)⟶{\mathrm{Hom}}_G(E{\otimes }_{F}V,E{\otimes }_{F}W),a\otimes f⟼((b\otimes v)\mapsto ab\otimes f(v))$ is an $E$-linear isomorphism.

Let $E\mathrm{/}F$ be finite Galois of characteristic $0$, let $G$ be finite, and let $V$ be a finite-dimensional $F$-representation. If $U$ is an irreducible constituent of $E{\otimes }_{F}V$, then every ${}^{\sigma }U$ is a constituent with the same multiplicity as $U$.

Let $F$ be characteristic $0$, $G$ finite, $E\mathrm{/}F$ finite Galois and a splitting field for $G$, and $V$ an irreducible $F$-representation. Then there are an absolutely irreducible constituent $U$ of $E{\otimes }_{F}V$ and an integer $m\ge 1$ such that $E{\otimes }_{F}V\cong m{⨁}_{\sigma \in \mathrm{Gal}(E\mathrm{/}F)\mathrm{/}\mathrm{Stab}(U)}{}^{\sigma }U\mathrm{.}$ The displayed summands are pairwise inequivalent.

Let $F\subseteq E\subseteq \mathbb{C}$ with $E\mathrm{/}F$ finite Galois, let $G$ be finite, and let $U$ be an absolutely irreducible $E$-representation with character $\chi$. Then $E^{\mathrm{Stab}(U)}=F(\chi )\mathrm{.}$

For an irreducible finite-dimensional $F$-representation $V$ of $G$, put $D_V:={\mathrm{End}}_{F[G]}(V)={\mathrm{End}}_G(V)\mathrm{.}$ Schur's lemma says that every nonzero member of this endomorphism ring is invertible, so $D_V$ is a division ring (Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and ${\mathrm{End}}_G(V)$ is a division ring). We call it the endomorphism division algebra of $V$. Scalars give a central embedding $F\hookrightarrow D_V$.

Let $F$ have characteristic $0$, $G$ be finite, and $V$ be an irreducible finite-dimensional $F$-representation. Then $V$ is absolutely irreducible if and only if $D_V=F$ (via scalar endomorphisms).

Let $\chi$ be an irreducible complex character of a finite group $G$, put $K=\mathbb{Q}(\chi )$, and choose a finite cyclotomic splitting field $E\mathrm{/}K$; thus $K\subseteq E\subseteq \mathbb{C}$ and $E\mathrm{/}K$ is finite Galois. Proposition 4.3.2 in the cited notes of Zheng says that, as $V$ ranges over the irreducible $K$-representations, the absolutely irreducible constituents of $E{\otimes }_{K}V$ form a complete, nonrepeating list of the irreducible $E$-representations, grouped into Galois orbits. Consequently there is a unique irreducible $K$-representation $V$, up to isomorphism, whose scalar extension contains a representation affording $\chi$. In the decomposition from Scalar extension of an irreducible finite-group representation, write its common multiplicity as $m$. The Schur index of $\chi$ over $K$ is $m_K(\chi ):=m\mathrm{.}$ The next lemma proves that enlarging the chosen finite Galois splitting field does not change this integer; that is why this is a definition rather than an auxiliary choice.

For an irreducible character $\chi$ of a finite group, the common multiplicity in the scalar-extension orbit used in The Schur index of an irreducible character is unchanged when the finite Galois splitting field is replaced by a larger finite Galois splitting field.

Let $F\subseteq E\subseteq \mathbb{C}$, let $V$ be an irreducible $F$-representation of a finite group $G$, and let $E\mathrm{/}F$ be finite Galois and a splitting field for $G$. Let $\chi$ be the character of one absolutely irreducible constituent of $E{\otimes }_{F}V$, let $m$ be that constituent's multiplicity in $E{\otimes }_{F}V$, and put $H=\mathrm{Stab}(\chi )$. Then ${\chi }_V=m{\sum }_{\sigma \in \mathrm{Gal}(E\mathrm{/}F)\mathrm{/}H}\sigma \chi \mathrm{.}$ In particular, ${\dim }_{F}V=m[F(\chi ):F]\chi (1)$.

Let $\chi$ be an irreducible complex character, put $K=\mathbb{Q}(\chi )$, and let $V$ be the irreducible $K$-representation used in The Schur index of an irreducible character. Then $m_K(\chi )\mid {\dim }_{K}V\mathrm{.}$

Let $D$ be a finite-dimensional division algebra whose centre is the field $K$. Then $[D:K]$ is a square. Its positive square root $\mathrm{ind}(D):=\sqrt{[D:K]}$ is the index (or degree) of $D$. Equivalently, every maximal subfield $L$ has $[L:K]=\mathrm{ind}(D)$ and splits $D$. This convention applies only to central finite-dimensional division algebras.

Let $\chi$ be an irreducible complex character of a finite group $G$, put $K=\mathbb{Q}(\chi )$, and let $V$ be the irreducible $K$-representation used in The Schur index of an irreducible character. If $D_V={\mathrm{End}}_G(V)$, then $Z(D_V)=K$ and $m_K(\chi )=\mathrm{ind}(D_V)\mathrm{.}$

Let $\chi$ be an irreducible complex character of a finite group and put $K=\mathbb{Q}(\chi )$. Its Schur index $m_K(\chi )$ is the least positive integer $r$ for which the character $r\chi$ is afforded by a $K$-representation. Consequently $\chi$ itself is realizable over $K$ if and only if $m_K(\chi )=1$.