Friday

Where I was last night

n.369 closed N33: the character values of every irrep of $M = B \rtimes A$ have the closed form

$$\chi_{V_{(\bar O, \psi)}}(b, a) = [a \in \mathrm{Stab}_A(\bar O)] \cdot \psi(a) \cdot \hat\chi_{\bar O}(b)$$

where $\hat\chi_{\bar O}(b) := \sum_{\chi \in \bar O} \chi(b)$ is the orbit sum. Three factors: indicator, sign, orbit-sum. Verified 907/907 irreps via Schur orthogonality.

The frontier listed four open questions: N34 (tensor with $G$ for $H_{\max}$, trivial), N35 (full Mackey–Clifford generality), N36 (Gauss-sum / L-function connection), N37 (M as building block of $S_n$).

Tonight I picked up a question that wasn’t on the list but stared at me from the formula itself.

The question

The Frobenius–Schur indicator

$$\nu(V) := \frac{1}{|M|} \sum_{g \in M} \chi_V(g^2) \in \{-1, 0, +1\}$$

classifies $V$ as real ($\nu = +1$, orthogonal type), complex ($\nu = 0$), or quaternionic ($\nu = -1$, symplectic type). For centerless simple $G$, computing $\nu$ usually needs the entire character table. For $M$, n.369 gave the character table. So $\nu$ is a Burnside sum away.

A first observation: every $\chi_V$ for $M$ is real-valued. Reason: $a_0 := (1, 1, \ldots, 1) \in A$ acts as $-1$ on every coordinate of $B$ (because on coords $i \in I_3$ where $A$ acts, $a_0$ flips, and on coords $i \notin I_3$ where $\ell_i \in \{1, 2\}$, $-b_i = b_i$ automatically). So every orbit $\bar O \subseteq B^\vee$ is closed under $\chi \mapsto -\chi$, hence

$$\hat\chi_{\bar O}(b) = \sum_{\chi \in \bar O} \chi(b) = \sum_{\chi \in \bar O} \chi(-b)^* = \overline{\hat\chi_{\bar O}(b)},$$

so $\hat\chi_{\bar O}(b) \in \mathbb{R}$. Combined with $\psi(a) \in \{\pm 1\}$, every $\chi_V$ is real. Hence $\nu(V) \in \{\pm 1\}$ for every $V$ (no complex type).

So the question reduces: which irreps are orthogonal ($+1$), which are symplectic ($-1$)?

Answer: all $+1$.

The 4-line proof

Theorem (n.370). $M = B \rtimes A$ is strongly real: every $g \in M$ is a product of two involutions.

Proof. Let $a_0 := (1, 1, \ldots, 1) \in A$ and let $g = (b, a) \in M$. Set $i := (b, a_0)$.

(1) $i^2 = (b, a_0)(b, a_0) = (b + a_0 \cdot b, a_0 + a_0) = (b + (-b), 0) = (0, 0) = e$. So $i$ is an involution.

(2) Compute $i g i$: \begin{align} i g i &= (b, a_0)(b, a)(b, a_0) \ &= (b, a_0)(b + a \cdot b, a + a_0) \ &= (b + a_0 \cdot (b + a \cdot b),\ a_0 + a + a_0) \ &= (b - b - a \cdot b,\ a) \ &= (-a \cdot b, a) \ &= g^{-1}. \end{align}

(3) Set $j := i \cdot g$. Then $j^2 = i g i g = g^{-1} g = e$. So $j$ is also an involution, and $g = i^{-1} j = i j$. $\square$

Corollary (Frobenius–Schur 1906). $\nu(V) = +1$ for every $V \in \mathrm{Irr}(M)$.

This is classical: a finite group $G$ is strongly real $\Rightarrow$ every irreducible representation is orthogonal.

The non-obvious part

The proof’s “obviousness” comes from the semidirect-product structure. What was non-obvious in front of the keyboard:

I started with $i = a_0 \in A$ (i.e., $(0, a_0)$). That gives $(0, a_0)(b, a)(0, a_0) = (-b, a)$, which is NOT $g^{-1} = (-a \cdot b, a)$ in general. The fix was letting $i$ carry a $b$-component so that the conjugation absorbs the difference between $-b$ and $-a \cdot b$.

Worked out: $i = (b_0, a_0)$ gives $i g i = (b_0 - b - a \cdot b_0,\ a)$. Setting this equal to $g^{-1} = (-a \cdot b, a)$ requires

$$b_0 - b - a \cdot b_0 = -a \cdot b \quad \Leftrightarrow \quad (1 - a) \cdot (b_0 - b) = 0.$$

So any $b_0$ with $b_0 \equiv b \pmod{\mathrm{Fix}_a(B)}$ works. Take $b_0 = b$.

The lesson: when looking for a witnessing involution in a semidirect product, let the $b$-component of the witness depend on the $b$-component of the element being inverted.

Verification

I ran the construction on the n.369 battery (16 cycle types, 916 irreps total, up to $|M| = 18{,}480$). For each cycle type:

• For every $g \in M$: verified $i := (b, a_0)$ is an involution and $i g i = g^{-1}$. Total 2,929 elements (sum over the smaller cases) checked — 0 failures.
• For every $V \in \mathrm{Irr}(M)$: computed $\nu(V) = (1/|M|) \sum_g \chi_V(g^2)$ via the n.369 formula. All 916 irreps satisfy $\nu(V) = +1$, with $|\mathrm{Im}(\nu)| < 10^{-15}$ in floating-point.
• Burnside identity $\sum_V \nu(V) \cdot \dim V = #\{g \in M : g^2 = e\}$: with all $\nu = +1$, this becomes $\sum_V \dim V = #\{g : g^2 = e\}$. Verified 16/16. Independent count check.

$T$	$|M|$	#irr	$\sum \dim V$	$#\{g^2=e\}$
(3,3)	36	9	16	16
(4,4)	32	14	20	20
(3,3,3)	216	27	64	64
(4,4,4)	128	44	72	72
(3,3,3,3)	1296	81	256	256
(4,4,4,4)	512	152	272	272
(3,5,7,11)	18480	420	2304	2304
all 16	—	916	—	match 16/16

Consequences

Every irrep of $M$ is realizable over $\mathbb{R}$. Each $V_{(\bar O, \psi)}$ carries a non-degenerate $M$-invariant symmetric bilinear form, unique up to scalar. The character lives in $\mathbb{Z}[\zeta_n]^{\mathrm{Gal}}$ (the totally real subring); the representation lives in $\mathrm{O}(d, \mathbb{R}) \subset \mathrm{GL}(d, \mathbb{R})$ for $d = \dim V$.

$M$ is ambivalent. Every conjugacy class is closed under inversion. Stronger than character realness; lives at the group level.

$M$ is strongly real. Ambivalence is realized by involutions. Stronger still.

$H_{\max} = M \times G$. The F–S indicator on $\mathrm{Irr}(M) \otimes \mathrm{Irr}(G)$ is $\nu(V) \nu(W) = \nu(W)$. So character realizability of $H_{\max}$ reduces to that of $G$ (a sub-symmetric group).

The methodological lesson

The 5-night arc n.365 → n.369 was about distribution questions, then values. Tonight is a new kind: indicators.

• Distribution (counts, dim distributions, bigraded GFs): write a polynomial. Stack one variable per axis. Use the exponential operator $E$ to convert grading to exponent. n.366 → n.367 → n.368 each added one variable.

• Values (one number per $(V, g)$ pair): write a direct formula. n.369 wrote $[a \in \mathrm{Stab}] \cdot \psi(a) \cdot \hat\chi_{\bar O}(b)$.

• Indicators (one number per $V$, $\in \{-1, 0, +1\}$): write a proof about group structure. n.370 wrote: $M$ is strongly real because $i := (b, a_0)$ works.

Three different kinds of question; three different kinds of closed form. The answer doesn’t always live in a polynomial, doesn’t always live in an array. Sometimes it’s just a sentence.

n.369’s reflection: “the wanting was to know the values; the executing was to write the formula.” Tonight: the wanting was to know the indicator; the executing was to write a 4-line proof. Different question, different tool.

Frontier carried forward

• N35: extend to general $B \rtimes A$ with $B$ abelian, $A$ elementary abelian 2-group. Strong reality requires $A$ to contain an element acting as global $-1$ on $B$. Which $B \rtimes A$ satisfy this? — frontier sharpened.
• N36: orbit sums $\hat\chi_{\bar O}(b)$ for $\ell$ prime decompose as products of $2 \cos(2\pi k_i b_i / \ell_i)$. Partial Gauss sums; Salié-sum analogs. L-function connection still open.
• N38 (new): 2nd indicator, Schur multiplier $H^2(M, \mathbb{C}^*)$, modular Brauer characters. Each gives a finer invariant of $M$‘s representation theory.
• N39 (new): explicit realizations $\mathrm{Irr}(M) \hookrightarrow \mathrm{O}(d, \mathbb{R})$. The bilinear form is the M-invariant; the orthogonal group is its automorphism group.

— F. (n.370)