Proposition. Schur elimination multiplies quadratic center characters [fdip-0044]
AGENTDRAFTED
For an enriched rational object, write its
relative precision in the specified orthogonal splitting as
\[Q=\begin {pmatrix}A&B\\B^\dagger &C\end {pmatrix},
\qquad S=A-BC^{-1}B^\dagger .\]
The positive elimination theorem gives invertible
positive \(C,S\). The adjoint uses the restricted reference
pairings \(g_{0,\mathrm {ret}},g_{0,\mathrm {el}}\).
Define the two rank-two forms
\[\begin {aligned}
q_S(x)&=g_{0,\mathrm {ret}}(x,Sx),\\
q_C(y)&=g_{0,\mathrm {el}}(y,Cy).
\end {aligned}\]
Their signed discriminants and associated quadratic Galois
characters satisfy
\[\delta (q)=\delta (q_S)\delta (q_C),\qquad
\chi _q=\chi _{q_S}\chi _{q_C}.\]
Multiplication of the \(\{1,-1\}\)-valued characters is their
tensor product. It retains the eliminated character as well
as the retained one.
The construction respects the declared arrows. If
\(T=T_{\mathrm {ret}}\oplus T_{\mathrm {el}}\) preserves the
pairing, action and splitting, the identity \(Q'T=TQ\) gives
\[\begin {aligned}
C'T_{\mathrm {el}}&=T_{\mathrm {el}}C,\\
B'T_{\mathrm {el}}&=T_{\mathrm {ret}}B,\\
(B')^\dagger T_{\mathrm {ret}}&=T_{\mathrm {el}}B^\dagger .
\end {aligned}\]
Inversion of the first relation and substitution in the
Schur formula yield
\(S'T_{\mathrm {ret}}=T_{\mathrm {ret}}S\).
The restricted maps consequently preserve \(q_C,q_S\) and
both reference pairings. The retained data inherit \(m^2\);
they carry no further splitting unless one is supplied.
Thus this naturality statement does not make elimination
an endofunctor on the already split rank-4 objects. A
retained trace ratio uses
\(L_{\mathrm {rel,ret}}=S-m^2I\), with the same trace formula.
The lower bound \(S\geq m^2I\) is inherited. Indeed, for
\(y_*=-C^{-1}B^\dagger x\), square completion and \(Q\geq m^2I\)
give
\[\begin {aligned}
g_{0,\mathrm {ret}}(x,Sx)&=q(x,y_*)\\
&\geq m^2\bigl (\|x\|^2+\|y_*\|^2\bigr )\\
&\geq m^2\|x\|^2,
\end {aligned}\]
where the norms use the restricted reference pairings. Thus
\(L_{\mathrm {rel,ret}}\geq 0\). Its trace must still be assumed
nonzero for the retained ratio to be defined. For example, with
counting pairing and the first summand retained,
\[Q=\operatorname {diag}(m^2I_2,(m^2+1)I_2)\]
has \(\operatorname {tr}(Q-m^2I)=2\), but \(S=m^2I_2\) and
\(L_{\mathrm {rel,ret}}=0\). The full trace-ratio domain is therefore
not inherited.
The triangular form isometry generally does not preserve
\(g_0\). Nor does its graph map
\[j_Q(x)=(x,-C^{-1}B^\dagger x)\]
preserve the reference pairing: its pulled-back value is
\[\begin {aligned}
g_0(j_Qx,j_Qx')&=g_{0,\mathrm {ret}}(x,x')\\
&\quad +g_{0,\mathrm {el}}(Kx,Kx').
\end {aligned}\]
It is a form isometry \(q(j_Qx)=q_S(x)\) into the full space,
not an enriched isomorphism. Clifford functoriality gives
an algebra embedding: after square completion it is the
inclusion of the Clifford algebra of an orthogonal summand,
injective on its monomial basis. But this embedding need
not carry even centers into the full even center. In an
orthogonal basis adapted to \(q_S\perp q_C\), the retained
volume \(e_1e_2\) anticommutes with the even element \(e_1f_1\),
where \(f_1\) is an eliminated nonisotropic generator.
Both products are nonzero, and characteristic is zero;
so this central element of the retained even algebra is
not central in the full even algebra.
Primitive Dirichlet representatives must be taken after
multiplying Galois characters. They are not in general the
pointwise product of the zero-extended functions on integers.
At \(t=1\), the cycle Schur computation has
\(\det Q=45\), \(\det S=5\) and \(C=3I_2\). The three signed
square classes are \([5],[-5],[-1]\), with fundamental
discriminants \(5,-20,-4\). The Galois product is \(\chi _5\),
but
\[\chi _{-20}(2)\chi _{-4}(2)=0,\qquad \chi _5(2)=-1.\]
The last value follows from
\(\chi _D(2)=(-1)^{(D^2-1)/8}\) for odd fundamental \(D\),
as in Brian Conrad's [conradQuadraticCharacters, Corollary 3.1, p. 4].
On integers coprime to 20 the product agrees with
\(\chi _5\); passing to its primitive representative restores
the prime-two value. No multiplication rule for the
corresponding L-functions follows from the character law.
Mathlib's DirichletCharacter.primitiveCharacter
and DirichletCharacter.primitiveCharacter_apply_of_isCoprime
separate primitive reduction from evaluation: agreement is
asserted at integers coprime to the original level, which
does not include the prime two in this example.