Return to Answer

The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. Thus, $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$ and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group. See below for a quick proof of this identity.

Of course, in $S_n$, all Frobenius-Schur indicators are 1, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to any group, for which any representation is either realisable over $\mathbb{R}$ or has non-real character, e.g. all abelian groups. Other cases would require more thought, but this is likely the right way to go about it.

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$ and the Frobenius-Schur indicator of $\chi$ is defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function. So let's just take inner products with all characters of $G$ to find their coefficients, noting that we can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta): $$ \begin{align*} \left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\ &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2) \end{align*}$$ as required.

Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1911 groups that have a symplectic representation, and for 1675 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that impose themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find an if and only if characterisation that provides more insight into the structure of the groups, whose square root counting functions is maximised by the identity? Following Pete's suggestion, I have started two follow-up questions on this business: one on square roots and one on $n$-th roots.

$\DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\SL}{SL}$ The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. For the definition, see the edit below. One has $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$, and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group. See below for a reference and a quick proof of this identity.

It follows from the usual theory of representations of $S_n$ that in this special case all Frobenius-Schur indicators are $1$, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to all groups for which every representation is either realisable over $\mathbb{R}$ or has non-real character, in other words has no symplectic (or sometimes called quaternionic) representations. That includes all abelian groups, all alternating groups, all dihedral groups, $\GL_n(\mathbb{F}_q)$ for all $n\in \mathbb{Z}_{\geq 1}$ and all prime powers $q$ (see [1, Ch. III, 12.6]), and many more.

[1] A. Zelevinsky, Representations of Finite Classical Groups, Lecture Notes in Mathematics, Vol. 869, Springer-Verlag, New York/Berlin, 1981.

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$, and $s(\chi)$ denotes the Frobenius-Schur indicator of $\chi$, defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function, so it is a linear combination of the irreducible characters of $G$. The coefficient of $\chi$ in this linear combination can be recovered as the inner products of $n$ with $\chi$. We can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta), so we obtain $$ \begin{align*} \left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\ &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2), \end{align*}$$ as claimed.

Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1911 groups that have a symplectic representation, and for 1675 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that suggest themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find a complete characterisation of the groups whose square root counting functions is maximised by the identity? Following Pete's suggestion, I have started two follow-up questions on this business: one on square roots and one on $n$-th roots.

The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. Thus, $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$ and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group. See below for a quick proof of this identity.

Of course, in $S_n$, all Frobenius-Schur indicators are 1, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to any group, for which any representation is either realisable over $\mathbb{R}$ or has non-real character, e.g. all abelian groups. Other cases would require more thought, but this is likely the right way to go about it.

Edit: One reference I have found for the identity expressing the number of square roots in terms of Frobenius-Schur indicators is Eugene Wigner, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 57-63, "On representations of certain finite groups". Once you get used to the notation, you will recognise it in displayed formula (11). Since the notation is really heavy going, I will supply a quick proof here:

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$ and the Frobenius-Schur indicator of $\chi$ is defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function. So let's just take inner products with all characters of $G$ to find their coefficients, noting that we can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta): $$ \begin{align*} \left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\ &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2) \end{align*}$$ as required.

Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1911 groups that have a symplectic representation, and for 1675 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that impose themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find an if and only if characterisation that provides more insight into the structure of the groups, whose square root counting functions is maximised by the identity? Following Pete's suggestion, I have started two follow-up questions on this business: one on square roots and one on $n$-th roots.

The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. Thus, $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$ and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group. See below for a quick proof of this identity.

Of course, in $S_n$, all Frobenius-Schur indicators are 1, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to any group, for which any representation is either realisable over $\mathbb{R}$ or has non-real character, e.g. all abelian groups. Other cases would require more thought, but this is likely the right way to go about it.

Edit: One reference I have found for the identity expressing the number of square roots in terms of Frobenius-Schur indicators is Eugene Wigner, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 57-63, "On representations of certain finite groups". Once you get used to the notation, you will recognise it in displayed formula (11). Since the notation is really heavy going, I will supply a quick proof here:

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$ and the Frobenius-Schur indicator of $\chi$ is defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function. So let's just take inner products with all characters of $G$ to find their coefficients, noting that we can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta): $$ \begin{align*} \left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\ &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2) \end{align*}$$ as required.

Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1911 groups that have a symplectic representation, and for 1675 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that impose themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find an if and only if characterisation that provides more insight into the structure of the groups, whose square root counting functions is maximised by the identity? Following Pete's suggestion, I have started two follow-up questions on this business: one on square roots and one on $n$-th roots.

The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. Thus, $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$ and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group. See below for a quick proof of this identity.

Of course, in $S_n$, all Frobenius-Schur indicators are 1, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to any group, for which any representation is either realisable over $\mathbb{R}$ or has non-real character, e.g. all abelian groups. Other cases would require more thought, but this is likely the right way to go about it.

Edit: One reference I have found for the identity expressing the number of square roots in terms of Frobenius-Schur indicators is Eugene Wigner, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 57-63, "On representations of certain finite groups". Once you get used to the notation, you will recognise it in displayed formula (11). Since the notation is really heavy going, I will supply a quick proof here:

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$ and the Frobenius-Schur indicator of $\chi$ is defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function. So let's just take inner products with all characters of $G$ to find their coefficients, noting that we can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta): $$ \begin{align*} \left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\ &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2) \end{align*}$$ as required.

Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1912 groups that have a symplectic representation, and for 1676 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that impose themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find an if and only if characterisation that provides more insight into the structure of the groups, whose square root counting functions is maximised by the identity? If anyone is interested in thinking about this some more, please let me know!

The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. Thus, $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$ and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to $$\sum_\chi s(\chi)\chi(g),$$ where the sum runs over all irreducible characters of the group. See below for a quick proof of this identity.

Of course, in $S_n$, all Frobenius-Schur indicators are 1, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to any group, for which any representation is either realisable over $\mathbb{R}$ or has non-real character, e.g. all abelian groups. Other cases would require more thought, but this is likely the right way to go about it.

Edit: One reference I have found for the identity expressing the number of square roots in terms of Frobenius-Schur indicators is Eugene Wigner, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 57-63, "On representations of certain finite groups". Once you get used to the notation, you will recognise it in displayed formula (11). Since the notation is really heavy going, I will supply a quick proof here:

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have $$n(g) = \sum_\chi s(\chi)\chi(g),$$ where the sum runs over all characters of $G$ and the Frobenius-Schur indicator of $\chi$ is defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function. So let's just take inner products with all characters of $G$ to find their coefficients, noting that we can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta): $$ \begin{align*} \left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\ &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2) \end{align*}$$ as required.

Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1911 groups that have a symplectic representation, and for 1675 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that impose themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find an if and only if characterisation that provides more insight into the structure of the groups, whose square root counting functions is maximised by the identity? Following Pete's suggestion, I have started two follow-up questions on this business: one on square roots and one on $n$-th roots.