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\begin{document}

\section{Polarization, vector invariants and symbolic
evaluation}\label{polarization-vector-invariants-and-symbolic-evaluation}

\emph{Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex at
Ultra. Original exposition, proofs, exercises and diagrams are CC0.}

Polarization moves an invariant between copies of a representation. We
prove how many copies suffice, obtain all vector and mixed-vector
invariants, and determine every bracket relation over an arbitrary
commutative ring. Tensor commutants explain why pairings and
determinants suffice. We then use these results to justify symbolic
evaluation and its completeness.

Let \(k\) have characteristic zero and \(V=k^n\). Copies of \(V\) are
rows \(v_i=(x_{i1},\ldots,x_{in})\); diagonal substitution commutes with
substitutions among the copies. Subgroups need not be reductive or
closed. \href{NOE-INV-01.html\#theorem-8-3}{Covariants, averaging and
Hilbert finiteness} supplies complete reducibility and the elementary
matrix-orbit argument. The integral bracket relation theorem states its
wider coefficient-ring scope separately.

\subsection{1. Polarizing and putting the variables back
together}\label{section-1}

Define \protect\phantomsection\label{equation-P1}{}

\[
D_{ij}=\sum_{a=1}^n x_{ia}\partial_{x_{ja}}\qquad(1\leq i,j\leq m).
\tag{P1}
\] If \(i\ne j\), Taylor's formula gives
\protect\phantomsection\label{equation-P2}{}

\[
F(\ldots,v_j+tv_i,\ldots)
=\sum_{r\geq0}\frac{t^r}{r!}D_{ij}^rF.
\tag{P2}
\] The sum is finite for each polynomial. For \(i=j\), \(D_{ii}\) is
instead the Euler operator measuring degree in row \(i\).

\protect\phantomsection\label{proposition-1-1}{}

\textbf{Proposition 1.1 (commutation).} The \(D_{ij}\) commute with
simultaneous linear substitution on \(V\), and
\protect\phantomsection\label{equation-P3}{}

\[
[D_{ij},D_{kl}]=\delta_{jk}D_{il}-\delta_{li}D_{kj}.
\tag{P3}
\]

\textbf{Proof.} Replacing the rows by \(gv_i\) and applying the chain
rule contracts the derivative with the same vector \(v_i\); the matrix
and its inverse cancel. This proves commutation with the diagonal
action, hence preservation of invariants. For (P3), apply the product
rule. The terms with two derivatives cancel in the difference.
Differentiating the multiplier \(x_{ka}\) gives
\(\delta_{jk}x_{ia}\partial_{x_{la}}\); differentiating \(x_{ia}\) in
the opposite product gives \(-\delta_{li}x_{ka}\partial_{x_{ja}}\). Sum
over \(a\). \(\square\)

A polynomial homogeneous of degree \(d\) on \(V\) has \textbf{full
polarization} \protect\phantomsection\label{equation-P4}{}

\[
\mathcal PF(v_1,\ldots,v_d)
=\frac1{d!}[t_1\cdots t_d]\,
F(t_1v_1+\cdots+t_dv_d).
\tag{P4}
\] It is symmetric and multilinear. \textbf{Restitution}, putting
\(v_1=\cdots=v_d=v\), recovers \(F(v)\): the coefficient of
\(t_1\cdots t_d\) in \((t_1+\cdots+t_d)^d\) is \(d!\). Conversely, a
symmetric multilinear form is determined by that diagonal polynomial,
since coefficient extraction in its diagonal expansion recovers \(d!\)
times the form. For a multihomogeneous polynomial, perform (P4)
separately in each row. All these operations commute with the diagonal
group action. Division by \(d!\) is the characteristic-zero step.

\subsection{2. The shifted determinant in Capelli's
identity}\label{section-2}

Noncommuting entries require an order. For an \(r\times r\) operator
matrix \(M\), put \[
\operatorname{cdet}M
=\sum_{\sigma\in S_r}\operatorname{sgn}(\sigma)
M_{\sigma(1),1}\cdots M_{\sigma(r),r}.
\] The rightmost operator acts first.

\protect\phantomsection\label{theorem-2-1}{}

\textbf{Theorem 2.1 (Capelli).} If \(m=n\) and \(X=(x_{ia})\), then
\protect\phantomsection\label{equation-P5}{}

\[
\operatorname{cdet}\bigl(D_{ij}+(n-j)\delta_{ij}\bigr)
=\det(X)\det(\partial_{x_{ia}}).
\tag{P5}
\] The shift can equally be written \(n-i\) on the diagonal. The stated
column order is essential.

\textbf{Proof.} Introduce anticommuting symbols \(e_1,\ldots,e_n\), with
\(e_i^2=0\), commuting with the ordinary variables and differential
operators. Put \[
\xi_a=\sum_i e_ix_{ia},\qquad
\eta_j=\sum_i e_iD_{ij}=\sum_a\xi_a\partial_{x_{ja}}.
\] For distinct row indices \(j_1,\ldots,j_r\), let \[
N_{j_1,\ldots,j_r}
=\sum_{a_1,\ldots,a_r}
\xi_{a_1}\cdots\xi_{a_r}
\partial_{x_{j_1a_1}}\cdots\partial_{x_{j_ra_r}};
\qquad N_\varnothing=1.
\] The derivatives in \(N\) stand to the right of all multipliers. The
product rule and \(\partial_{x_{ja}}\xi_b=\delta_{ab}e_j\) give
\protect\phantomsection\label{equation-P6}{}

\[
\eta_jN_{j_1,\ldots,j_r}
=N_{j,j_1,\ldots,j_r}-r e_jN_{j_1,\ldots,j_r}.
\tag{P6}
\] To check the sign, when the derivative hits the \(l\)-th multiplier
\(\xi_{a_l}\), it yields \(\xi_{a_l}\xi_{a_1}\cdots\xi_{a_{l-1}}e_j
\xi_{a_{l+1}}\cdots\xi_{a_r}\). Moving \(e_j\) to the front gives sign
\((-1)^l\); moving \(\xi_{a_l}\) to its original position gives
\((-1)^{l-1}\). Their product is \(-1\). Each of the \(r\) terms
therefore contributes \(-e_jN\).

Starting from the right and using (P6),
\protect\phantomsection\label{equation-P7}{}

\[
(\eta_1+(n-1)e_1)(\eta_2+(n-2)e_2)\cdots\eta_n
=N_{1,\ldots,n}.
\tag{P7}
\] The coefficient of \(e_1\cdots e_n\) on the left is the column
determinant in (P5). On the right, \[
\xi_{a_1}\cdots\xi_{a_n}
=e_1\cdots e_n\det(x_{i,a_j})_{i,j}.
\] Only tuples that permute \(1,\ldots,n\) remain nonzero. Their signed
derivative sum is \(\det(\partial_{x_{ia}})\); their common multiplier
is \(\det X\). Taking the coefficient proves (P5). \(\square\)

For two rows, the identity reads
\protect\phantomsection\label{equation-P8}{}

\[
(D_{11}+1)D_{22}-D_{21}D_{12}
=(x_{11}x_{22}-x_{12}x_{21})
(\partial_{11}\partial_{22}-\partial_{12}\partial_{21}).
\tag{P8}
\] The added \(D_{22}\) cancels the term created when a derivative in
\(D_{21}\) hits a multiplier in \(D_{12}\). Formula (P6) is that same
cancellation for all sizes.

\subsection{3. A row bound without an unproved highest-weight
theorem}\label{section-3}

Write \(H_d=k [V^m]_d\). A subspace stable under row substitutions is
stable under all \(D_{ij}\). Conversely, stability under all \(D_{ij}\)
implies stability under row substitutions. For off-diagonal elementary
matrices this follows from (P2). For diagonal substitutions, the
commuting \(D_{ii}\) have nonnegative integer eigenvalues; projections
onto their eigenspaces are interpolation polynomials in them. Thus a
stable subspace is multihomogeneous and stable under row scalings.
Elementary matrices and diagonal matrices generate
\(\operatorname{GL}_m(k)\).

\protect\phantomsection\label{lemma-3-1}{}

\textbf{Lemma 3.1 (complete reducibility for row substitutions).} Every
\(\operatorname{GL}_m(k)\)-stable subspace of \(H_d\) has a
\(\operatorname{GL}_m(k)\)-stable complement.

\textbf{Proof.} Over \(\mathbb C\), apply complete reducibility for
\(\operatorname{SL}_m\) from Theorem 8.3 of the first lesson. Scalar
matrices act on \(H_d\) by scalars. Every complex general-linear matrix
is a scalar times a special-linear matrix, so that complement is
general-linear stable.

For arbitrary \(k\), let \(M\subset H_d\) be stable and choose a
\(k\)-basis. Its finitely many entries in the monomial basis lie in a
finitely generated field \(L\subset k\) over \(\mathbb Q\). The matrices
of \(D_{ij}\) in the monomial basis are rational. Coordinates expressing
\(D_{ij}\) of a chosen basis vector in the chosen basis can be computed
using one nonzero maximal basis minor, so belong to \(L\). Hence the
\(L\)-span \(M_L\) is stable.

Embed \(L\) into \(\mathbb C\): choose algebraically independent complex
values for a transcendence basis and then extend over the finite
algebraic extension into the algebraically closed field \(\mathbb C\).
Complex complete reducibility gives a projection
\(p:H_d\otimes_L\mathbb C\to M_L\otimes_L\mathbb C\), commuting with
every \(D_{ij}\) and equal to the identity on \(M_L\). Those properties
are a finite system of linear equations for the entries of \(p\), with
coefficients in \(L\). Gaussian elimination shows that a system
consistent over \(\mathbb C\) is consistent over \(L\). Extend an
\(L\)-solution to \(k\). Its kernel is a \(D_{ij}\)-stable, hence
\(\operatorname{GL}_m(k)\)-stable, complement. \(\square\)

\protect\phantomsection\label{lemma-3-2}{}

\textbf{Lemma 3.2 (a highest row weight uses at most \(n\) rows).} A
nonzero finite-dimensional \(\operatorname{GL}_m(k)\)-stable subspace of
\(H_d\) contains a nonzero multihomogeneous polynomial \(F\) killed by
every \(D_{ij}\) with \(i<j\). If \(m\geq n\), every such \(F\) depends
only on its first \(n\) rows.

\textbf{Proof.} Decompose into row multidegrees
\(\lambda=(\lambda_1,\ldots,\lambda_m)\), using the interpolation
projectors just described. Choose the lexicographically largest weight
with a nonzero vector \(F\). For \(i<j\), \(D_{ij}\) changes that weight
to \(\lambda+e_i-e_j\), a larger weight, or gives zero. Maximality
therefore forces zero.

For \(j>n\), the equations \(D_{ij}F=0\), \(1\leq i\leq n\), are \[
\sum_{a=1}^n x_{ia}\partial_{x_{ja}}F=0.
\] In the fraction field of the polynomial ring the first \(n\)-row
matrix is invertible. Thus every \(\partial_{x_{ja}}F=0\). These
derivatives are polynomial and hence zero before localization. In
characteristic zero, zero derivative means independence of that
variable. This holds for every \(j>n\). \(\square\)

The phrase ``highest weight'' here means precisely the maximal
multidegree and its annihilation conditions. No classification of
irreducible representations was used.

\protect\phantomsection\label{theorem-3-3}{}

\textbf{Theorem 3.3 (Weyl's polarization theorem).} For every subgroup
\(G\subset\operatorname{GL}(V)\) and \(m\geq n\),
\protect\phantomsection\label{equation-P9}{}

\[
k [V^m]^G
=\operatorname{span}_k\{\text{row substitutions of }F:
F\in k [V^n]^G\}.
\tag{P9}
\] If a set generates the smaller invariant ring as an algebra, its
polarizations generate the larger invariant ring.

\textbf{Proof.} In each total degree the invariant subspace \(M=H_d^G\)
is stable under \(\operatorname{GL}_m\), by Proposition 1.1. Split it
into simple submodules using Lemma 3.1. Every nonzero simple summand
contains a highest-row-weight vector by Lemma 3.2. That vector is
invariant and belongs to \(k [V^n]\). Its orbit spans the simple
summand, since its span is a nonzero stable subspace. Summing proves
(P9).

Equation (P2), coefficient interpolation in \(t\), and the diagonal
interpolation projectors show that orbit span and the span of all
iterated \(D_{ij}\) agree. If \(S\) generates \(k [V^n]^G\), the algebra
generated by its row translates contains all translates of every
polynomial in \(S\), because row substitution is an algebra
homomorphism. It therefore contains the space in (P9). Equivalently, the
Leibniz rule shows that all polarizations of those polynomials belong to
the algebra generated by polarizations of \(S\). \(\square\)

For \(m<n\), restrict the invariant ring on \(n\) copies by setting the
extra rows to zero. Inclusion followed by restriction is the identity on
the smaller invariant ring, so restrictions of its generators suffice.

\subsection{4. Brackets generate the special-linear
invariants}\label{section-4}

\protect\phantomsection\label{theorem-4-1}{}

\textbf{Theorem 4.1 (first fundamental theorem for vectors).} For any
\(m\), the invariant ring \(k [V^m]^{\operatorname{SL}_n}\) is generated
by the \(n\)-row brackets \[
[i_1\cdots i_n]=\det(v_{i_1},\ldots,v_{i_n}).
\] When \(m<n\), this ring is \(k\).

\textbf{Proof.} The matrix-orbit proof in Section 7 of the first lesson
gives \(k [V^n]^{\operatorname{SL}_n}=k [\det(v_1,\ldots,v_n)]\). That
proof works over any infinite field: invertible matrices of a specified
determinant form a single special-linear orbit, and a polynomial
equality on the open invertible locus is global.

Apply Theorem 3.3. A row substitution replaces the determinant by the
determinant of \(n\) linear combinations of the rows. By multilinearity
and alternation it is a linear combination of \(n\)-row brackets. A
translate of a polynomial in the determinant is therefore a polynomial
in these brackets, proving generation. If \(m<n\), the restrictions of
all such brackets are zero; the restriction argument after Theorem 3.3
leaves only constants. \(\square\)

For three binary vectors, there are three brackets \([12],[13],[23]\)
and no relation. On the open set \([12]=p\ne0\), choose \[
v_1=(1,0),\quad v_2=(0,p),\quad
v_3=(-r/p,q).
\] Then the three brackets are \(p,q,r\). Their values fill a dense open
set of affine three-space; thus a polynomial relation between them is
zero.

For four binary vectors, expansion gives
\protect\phantomsection\label{equation-P10}{}

\[
[12][34]-[13][24]+[14][23]=0.
\tag{P10}
\] It is the only relation. In the abstract ring on six bracket
variables modulo (P10), the relation is primitive and linear in
\([34]\); its leading coefficient \([12]\) is coprime to its constant
coefficient. Gauss's lemma makes that quotient a domain. After inverting
\([12]\), the remaining five brackets are independent: use the displayed
coordinates for \(v_3\) and the corresponding pair of free coordinates
for \(v_4\). The map from the abstract quotient to the actual invariant
ring is therefore injective after localization, and hence injective
before localization because the quotient is a domain. Theorem 4.1 gives
surjectivity.

\subsubsection{All bracket relations, with integer
coefficients}\label{all-bracket-relations-with-integer-coefficients}

Generation and relations are different questions. We now give a
presentation of the bracket algebra that also keeps track of its
integral coefficient lattice. In this subsection \(R\) can be any
commutative ring. Use an \(n\) by \(m\) matrix of independent variables
\(z_{ai}\); its \(i\)-th column represents the vector previously written
as row \(v_i\).

For every strictly increasing \(n\)-tuple \(I\) introduce an
indeterminate \(p_I\). Extend the notation to an ordered tuple by
alternation, with a repeated index giving zero. A product
\(p_{I_1}\cdots p_{I_s}\) is \textbf{standard} if \[
(I_j)_a\leq (I_{j+1})_a
\quad\text{for every }a=1,\ldots,n,\ j=1,\ldots,s-1.
\] Thus its columns increase strictly downward and its rows increase
weakly to the right. The empty product is standard.

Here are the quadratic Grassmann--Plücker shuffle relations. Take
ordered lists \(H,T,S\) of respective lengths \(t-1,n-t,n+1\), with
\(1\leq t\leq n\), and set \protect\phantomsection\label{equation-P21}{}

\[
\sum_{\substack{C\subset\{1,\ldots,n+1\}\\|C|=n-t+1}}
\epsilon(C)\,p_{H,S_C}\,p_{S_{C^c},T}=0.
\tag{P21}
\] The sublists retain their order in \(S\), and \(\epsilon(C)\) is the
sign of the permutation listing \(C\) first and \(C^c\) second. Signed
tuples in (P21) are interpreted by the alternating convention. For
\(n=2\) the nonzero relations are exactly (P10), with arbitrary four
distinct indices.

\protect\phantomsection\label{theorem-4-2}{}

\textbf{Theorem 4.2 (integral second fundamental theorem).} The
homomorphism \[
R [p_I]\longrightarrow R [z_{ai}],\qquad
p_I\longmapsto \det(z_{a,i_b})_{a,b=1}^n
\] has kernel generated by (P21). Its image is a free \(R\)-module with
the standard bracket products as a basis. In particular the presentation
and the basis remain valid modulo every prime, and no additional
relation appears there.

\textbf{Proof.} First, (P21) vanishes on actual determinants. Its left
side is an alternating multilinear function of the \(n+1\) columns
listed in \(S\). Indeed, an exchange within a chosen sublist changes its
determinant sign; an exchange between the two sublists pairs the
corresponding subset terms with opposite signs. It is zero when two
columns coincide, including in characteristic two. Expand all columns in
the \(n\) coordinate basis vectors. Every term has two equal basis
columns, so the alternating function is zero. This argument is integral
and uses no division.

Next we straighten a nonstandard product. Order its increasing column
tuples lexicographically. If the product is not standard, two
consecutive columns \(A=(a_1,\ldots,a_n)\), \(B=(b_1,\ldots,b_n)\) have
\(a_t>b_t\) for some position \(t\); take the first such position. Put
\[
H=(a_1,\ldots,a_{t-1}),\quad
T=(b_{t+1},\ldots,b_n),\quad
S=(a_t,\ldots,a_n,b_1,\ldots,b_t).
\] The first \(n-t+1\) entries of \(S\) are all larger than its last
\(t\) entries, because \(a_t>b_t\). The term choosing those first
entries in (P21) is \(p_Ap_B\), with coefficient one. Every other
nonzero term chooses at least one \(b_i\leq b_t<a_t\) for its first
determinant. After sorting that determinant, its tuple is
lexicographically smaller than \(A\): the fixed initial entries \(H\)
are retained, and at least one of the larger tail entries has been
replaced by a smaller entry.

No other term is the original product with its factors reversed. To see
this, \(A<B\) lexicographically, since equal columns cannot violate
componentwise order. Their first unequal position occurs before or at
\(t\), and cannot be \(t\), where \(a_t>b_t\). Its entry in \(A\) is
smaller than the corresponding entry in \(B\), and is not an entry of
\(B\): all earlier entries agree and all later entries of \(B\) are
larger. That entry belongs to \(H\), so the first determinant in another
term cannot be \(p_B\).

Consequently (P21) expresses the original product as an integer linear
combination of products whose sorted list of column tuples is
lexicographically smaller. Other factors do not affect this conclusion:
inserting the smaller new column before the former column \(A\) strictly
decreases the sorted list. The number of columns and the multiset of all
their indices are preserved. There are only finitely many such lists, so
repeated straightening terminates in standard products. This proves that
standard products span the quotient by the shuffle relations over \(R\).

They are independent after substitution. Order the matrix variables
lexicographically, row by row: \[
z_{11}>z_{12}>\cdots>z_{1m}>
z_{21}>\cdots>z_{nm}.
\] The leading monomial of an increasing minor is its diagonal monomial
\(\prod_a z_{a,i_a}\), with coefficient one: the determinant term using
the smallest column in row one is largest, then the same argument
applies to subsequent rows. Thus a standard product has leading monomial
\[
\prod_{j=1}^s\prod_{a=1}^n z_{a,(I_j)_a}.
\] This monomial determines the standard product. For each coordinate
row \(a\), it records the multiset of entries in row \(a\) of the
tableau; weak increase puts that multiset in its unique order,
recovering every column. Distinct standard products therefore have
distinct leading monomials.

In a nonzero finite \(R\)-linear combination of their images, choose the
largest leading monomial among terms with nonzero coefficients.
Smaller-leading polynomials cannot contribute to it, and its coefficient
is the chosen nonzero coefficient times one. The combination is nonzero,
even when \(R\) has zero divisors. Independence follows. Spanning and
independence identify the kernel exactly with the shuffle ideal and
establish all assertions. \(\square\)

Combining Theorems 4.1 and 4.2 gives the full presentation of the vector
invariant ring in characteristic zero. The relation theorem itself has
no characteristic restriction. It also shows why a computation after
inverting one bracket alone would not suffice: the standard basis
controls the entire polynomial algebra, including its specializations at
that bracket's zero set.

\protect\phantomsection\label{corollary-4-3}{}

\textbf{Corollary 4.3 (integral vector invariants).} A polynomial with
integer coefficients, universally invariant under
\(\operatorname{SL}_n\) on any number of vector columns, is an integer
polynomial in their brackets.

\textbf{Proof.} Over \(\mathbb Q\), Theorem 4.1 expresses it in
brackets. Theorem 4.2 straightens that expression into a rational linear
combination of standard products. Clear denominators with the least
positive integer \(N\), obtaining \[
NF=\sum_T c_T p_T,\qquad c_T\in\mathbb Z.
\] If a prime \(p\) divides \(N\), reduction modulo \(p\) makes the left
side zero. Independence of standard products over \(\mathbb F_p\) makes
every \(c_T\) divisible by \(p\). Dividing the integral identity by
\(p\) contradicts minimality of \(N\). Hence \(N=1\). Conversely
determinants are universally special-linear invariant, so all integer
polynomials in them are invariant. \(\square\)

Noether's 1919 paper, Section 3, proves precisely the binary integral
statement by normalizing bracket products and controlling relations
modulo each prime. The tableau argument here supplies the same integral
control in every dimension. Its binary step is particularly concrete: \[
p_{14}p_{23}=p_{13}p_{24}-p_{12}p_{34}.
\] The nested pair on the left is replaced by the two standard pairings
on the right. Leading diagonal monomials then certify that the standard
expressions are unique.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={Three pairings of four vector indices. The nested product of brackets 14 and 23 equals the product 13 and 24 minus the product 12 and 34. The two resulting products have componentwise ordered columns.}]{figures/bracket-straightening.png}}
\caption{Three pairings of four vector indices. The nested product of
brackets 14 and 23 equals the product 13 and 24 minus the product 12 and
34. The two resulting products have componentwise ordered columns.}
\end{figure}

The arcs record which indices belong to each bracket; they do not
represent a geometric configuration of the vectors. The signs are those
of (P10). This is the \(n=2\) straightening step in Theorem 4.2, with no
denominator introduced. Editable diagram.

\subsection{5. Repeated kinds of ground forms}\label{section-5}

Let \(U\) be a representation of dimension \(D\), and let \(Z\) be
another representation containing all the other ground forms. A kind of
form occurring \(N\) times has coefficient space \(U^N\). The group acts
diagonally on those copies and on \(Z\).

\protect\phantomsection\label{theorem-5-1}{}

\textbf{Theorem 5.1 (Noether's reduction for repeated kinds).} In
characteristic zero, \[
k [Z\oplus U^N]^G
\] is generated by polarizations, among the \(U\)-copies, of the
invariants on \(Z\oplus U^{\min(N,D)}\). The operators do not act on the
other kinds of ground forms.

\textbf{Proof.} Repeat Lemmas 3.1--3.2 in a component of fixed total
\(U\)-degree and fixed \(Z\)-degree. It is the tensor product of
\(k [U^N]_d\) with the finite-dimensional space \(k [Z]_e\), on which
the row-substitution group acts trivially. The same
complete-reducibility proof applies. For a highest-row-weight invariant,
the derivative equations in a row \(j>D\) are a \(D\times D\) system
with the first \(D\) coefficient vectors as its matrix. In their
fraction field that matrix is invertible, so the polynomial is
independent of every later \(U\)-row. The orbit-span argument proves the
assertion for \(N\geq D\). If \(N<D\), there is no reduction to perform.
\(\square\)

For binary quadratics \(D=3\), not two. Noether's 1916 paper credits the
underlying reduction of repeated vector rows to Capelli, Mertens and
Deruyts. Her answer to Hilbert applies that reduction to coefficient
vectors of repeated ground forms. The number of copies is bounded by the
dimension of the coefficient representation, not by the number of
original polynomial variables.

\subsection{6. Vectors and dual vectors}\label{section-6}

The symbolic method also needs invariants involving both vectors and
covectors. We give the tensor argument rather than assume a mixed
version of the vector theorem.

\protect\phantomsection\label{lemma-6-1}{}

\textbf{Lemma 6.1 (a tensor commutant).} The endomorphisms of
\(V^{\otimes r}\) commuting with every \(g^{\otimes r}\),
\(g\in\operatorname{GL}(V)\), are spanned by permutations of the \(r\)
tensor factors.

\textbf{Proof.} First work over \(\mathbb C\). Let
\(E=\operatorname{End}(V)\). Under
\(\operatorname{End}(V^{\otimes r})=E^{\otimes r}\), conjugation by a
factor permutation permutes the \(r\) factors. Its fixed subspace is the
symmetric tensors, spanned by \(T^{\otimes r}\) with \(T\in E\). To
verify the spanning assertion, polarize the power
\((t_1T_1+\cdots+t_rT_r)^{\otimes r}\); its \(t_1\cdots t_r\)
coefficient is the sum over factor permutations. Averaging factor
permutations spans the fixed subspace in characteristic zero. Invertible
\(T\) suffice: a linear functional vanishing on all their powers gives a
polynomial on \(E\) vanishing on the dense invertible open set, and is
therefore zero on every power.

Thus the linear span of \(g^{\otimes r}\) is the commutant of the
permutation algebra \(P\). We need that the double commutant of \(P\) is
\(P\). Here is the finite-dimensional argument. Averaging over the
finite group \(S_r\) makes every submodule split: average any linear
projection onto the submodule, and its kernel is an invariant
complement. Decompose \(V^{\otimes r}\) into copies of pairwise
nonisomorphic simple \(P\)-modules \(U_a\). An endomorphism of a simple
module is scalar: over \(\mathbb C\) it has an eigenvalue, and the
kernel of the corresponding difference from a scalar is a nonzero
submodule, hence the whole module. A map between distinct simples is
zero, by its kernel and image.

The image of \(P\) is the full product of matrix algebras on the
\(U_a\). To prove this assertion, choose a basis
\(u_{a,1},\ldots,u_{a,d_a}\) of each \(U_a\), and consider the image \[
M=\{(pu_{a,j})_{a,j}:p\in P\}
\subset\bigoplus_aU_a^{d_a}.
\] If it were proper, its split quotient would admit a nonzero map to
some \(U_b\). Composing with the quotient gives a nonzero \(P\)-linear
map annihilating \(M\). The preceding simple-module facts make that map
\(\sum_jc_j\) times the coordinate identity on the copies of \(U_b\),
and zero on other types. Evaluating at \(p=1\) yields
\(\sum_jc_ju_{b,j}=0\), contradicting basis independence. Hence \(M\) is
the whole direct sum. Specifying arbitrary images of those bases proves
the assertion.

Writing the original space as \(\bigoplus_a U_a\otimes\mathbb C^{m_a}\),
its commutant is therefore
\(\bigoplus_a I_{U_a}\otimes\operatorname{End}(\mathbb C^{m_a})\). The
commutant of this algebra is
\(\bigoplus_a\operatorname{End}(U_a)\otimes I\), precisely the image of
\(P\). This proves the double-commutant assertion and the lemma over
\(\mathbb C\).

Over any characteristic-zero \(k\), commuting with all \(g^{\otimes r}\)
is a system of rational linear equations: expand the commutator as a
polynomial in the entries of a generic matrix. Vanishing for invertible
matrices over the infinite field is equivalent to vanishing of all
coefficients. The permutation operators have rational entries, and
Gaussian elimination shows that equality of the two solution spaces over
\(\mathbb C\) implies equality over \(k\). \(\square\)

\protect\phantomsection\label{theorem-6-2}{}

\textbf{Theorem 6.2 (mixed first fundamental theorem).} The
special-linear invariant ring on any number of vectors \(v_i\) and dual
vectors \(\alpha_j\) is generated by their pairings \(\alpha_j(v_i)\),
vector determinants and dual-vector determinants.

\textbf{Proof.} Full polarization from (P4) reduces generation to
multilinear invariants in \(r\) vector slots and \(s\) dual slots. First
consider the general linear group. Scalar matrices make such an
invariant zero unless \(r=s\). When \(r=s\), it corresponds to an
endomorphism of \(V^{\otimes r}\); invariance says that endomorphism
commutes with all \(g^{\otimes r}\). Lemma 6.1 expresses it as a sum of
factor permutations. Each corresponding multilinear polynomial is
\(\prod_j\alpha_j(v_{\sigma(j)})\). This proves the mixed general-linear
theorem.

For the special linear group, first work over an algebraic closure. The
central matrices \(\zeta I\), \(\zeta^n=1\), show that \(r-s=qn\) for an
integer \(q\), unless the invariant is zero. A nonzero multilinear
invariant \(F\) has general-linear determinant character \(q\): write
\(g=\lambda h\), \(\lambda^n=\det g\), \(h\in\operatorname{SL}_n\), and
use multilinearity to get \(F(gv,\alpha g^{-1})=\det(g)^qF(v,\alpha)\).

If \(q\geq0\), introduce \(q\) blocks of \(n\) auxiliary dual vectors
\(\beta_{\ell,1},\ldots,\beta_{\ell,n}\). The product \[
F\prod_{\ell=1}^q\det(\beta_{\ell,1},\ldots,\beta_{\ell,n})
\] is general-linear invariant and multilinear in all its slots. The
general-linear theorem writes it as a sum of products of pairings.
Antisymmetrize in each auxiliary block, dividing by \(n!\). The left
side is unchanged. On the right, the \(n\) pairings using one block
become \[
\frac1{n!}\det(\beta_{\ell,1},\ldots,\beta_{\ell,n})
\det(v_{i_1},\ldots,v_{i_n}).
\] If a vector slot repeats there, its determinant is zero. Cancel the
product of auxiliary determinants in the polynomial domain. The result
expresses \(F\) in vector determinants and the original pairings. If
\(q<0\), use auxiliary vector blocks instead and obtain dual
determinants and pairings.

Finally restitute the polarized slots. Descent to \(k\) is again a
finite rational linear-algebra assertion in each degree: invariance is
the kernel of the coefficient equations for elementary unipotent
substitutions, and the proposed spanning polynomials have integer
coefficients. Gaussian elimination descends their spanning equality.
\(\square\)

This mixed theorem is the input for symbolic completeness in Section 9
below.

\subsubsection{Two commuting substitution
groups}\label{two-commuting-substitution-groups}

The same commutant argument explains the duality viewpoint on matrix
polynomials. We state it over \(\mathbb C\), so simple finite-group
modules and their multiplicity spaces can be used without a
splitting-field qualification. Let \(U=\mathbb C^m\), \(V=\mathbb C^n\).
Homogeneous degree-\(d\) polynomials in a matrix have the representation
\(\operatorname{Sym}^d(U\otimes V)\), with duals inserted if the inverse
action on coordinate functions is chosen. Row and column substitutions
commute.

\protect\phantomsection\label{proposition-6-3}{}

\textbf{Proposition 6.3 (matrix duality).} On each degree piece, the
linear spans of the two general-linear substitution groups are mutual
commutants. The joint representation is a direct sum of pairwise
inequivalent simple tensor-product representations, each occurring once.

\textbf{Proof.} Averaging over \(S_d\) identifies the symmetric power
with \[
\bigl(U^{\otimes d}\otimes V^{\otimes d}\bigr)^{S_d},
\] where the same permutation acts on both tensor powers. Reorder the
tensor factors to obtain this identification from
\((U\otimes V)^{\otimes d}\).

Let \(S_a\) range over the simple complex \(S_d\)-modules. The
finite-group splitting and simple-module arguments proved in Lemma 6.1
give \[
U^{\otimes d}=\bigoplus_a M_a(U)\otimes S_a,\qquad
V^{\otimes d}=\bigoplus_b M_b(V)\otimes S_b,
\] where a multiplicity space can be defined as
\(M_a(U)=\operatorname{Hom}_{S_d}(S_a,U^{\otimes d})\). Absent types
have zero multiplicity space. That lemma also proves that the span of
\(\operatorname{GL}(U)\) on its tensor power is the full commutant of
the permutation algebra. Hence its action on the nonzero \(M_a(U)\) has
image \(\prod_a\operatorname{End}(M_a(U))\); the analogous statement
holds for \(V\). Thus these are simple, pairwise inequivalent
general-linear modules. Simplicity follows from the full matrix image,
and inequivalence from the ability to act independently on distinct
factors.

Now \[
(S_a\otimes S_b)^{S_d}
\cong\operatorname{Hom}_{S_d}(S_a^*,S_b).
\] Duals of simple finite-group modules are simple: the annihilator of a
proper dual submodule would be an invariant subspace of the original
module. Schur's lemma, proved in Lemma 6.1, makes the displayed space
zero unless \(S_b\cong S_a^*\), when it is one-dimensional. If \(a^*\)
denotes that dual type, the degree piece is therefore \[
\bigoplus_a M_a(U)\otimes M_{a^*}(V),
\] omitting zero terms.

The two group spans on this direct sum are respectively
\(\prod_a\operatorname{End}(M_a(U))\otimes I\) and
\(\prod_a I\otimes\operatorname{End}(M_{a^*}(V))\). Their commutants are
each other, by the same matrix-block computation used in Lemma 6.1. Each
joint tensor summand is simple because the joint algebra contains its
full endomorphism algebra, and different summands are inequivalent
because their first factors are. Each occurs once in the displayed
decomposition. This proves the proposition without an unproved
classification of highest weights. \(\square\)

This is the polynomial matrix duality associated with the modern
viewpoint of Howe; see his
\href{https://www.math.utah.edu/~ptrapa/math-library/howe/howe-remarks-invariant-theory-TAMS-1989.pdf}{Remarks
on classical invariant theory}, especially its introduction on the
relation between polynomial and tensor invariants. A partition
description of the multiplicity spaces gives more names for the
summands, but is not needed for the polarization bound or for the
mutual-commutant assertion proved here.

\subsection{7. The binary Clebsch--Gordan expansion}\label{section-7}

Use two binary pairs \(x=(x_1,x_2)\), \(y=(y_1,y_2)\). Put \[
B=[xy]=x_1y_2-x_2y_1,\qquad
\Omega=\partial_{x_1}\partial_{y_2}
-\partial_{x_2}\partial_{y_1}.
\] For a binary form \(h(z)\) of degree \(a+b\), define its normalized
\((a,b)\)-polar by \protect\phantomsection\label{equation-P11}{}

\[
\mathcal P_{a,b}h(x,y)
=\frac1{\binom{a+b}{b}}[t^b]h(x+ty).
\tag{P11}
\] It has bidegree \((a,b)\), equals \(h(x)\) on \(y=x\), and is killed
by \(\Omega\). The last assertion follows by expanding the derivatives
of \(h\): its second derivatives are symmetric, so the alternating
contraction is zero.

\protect\phantomsection\label{theorem-7-1}{}

\textbf{Theorem 7.1 (binary series, with coefficients).} If \(F(x,y)\)
has bidegree \((p,q)\), let \[
h_k(z)=\left.\Omega^kF(x,y)\right|_{x=y=z},
\quad 0\leq k\leq\min(p,q).
\] Then \protect\phantomsection\label{equation-P12}{}

\[
F(x,y)=\sum_{k=0}^{\min(p,q)}
\frac{(p+q-2k+1)!}{k!(p+q-k+1)!}
B^k\,\mathcal P_{p-k,q-k}h_k(x,y).
\tag{P12}
\] All operators here are equivariant for \(\operatorname{SL}_2\); the
expansion also applies with other coefficient variables held fixed.

\textbf{Proof.} The product rule gives, for a bihomogeneous \(L\) of
total degree \(N\), \[
\Omega(BL)=B\Omega L+(N+2)L.
\] If \(\Omega L=0\), induction therefore gives \[
\Omega(B^kL)=k(N+k+1)B^{k-1}L.
\] Apply this \(k\) times: \protect\phantomsection\label{equation-P13}{}

\[
\left.\Omega^k(B^kL)\right|_{y=x}
=k!\frac{(N+k+1)!}{(N+1)!}L(x,x).
\tag{P13}
\] Fewer than \(k\) contractions leave a factor \(B\), which is zero on
the diagonal. More than \(k\) contractions kill the remaining harmonic
\(L\).

The maps \(h\mapsto B^k\mathcal P_{p-k,q-k}h\), with \(h\) of degree
\(p+q-2k\), have linearly independent images: apply \(\Omega^k\) and
then diagonal restriction, which isolates that summand by (P13). The sum
of their dimensions is \[
\sum_{k=0}^{\min(p,q)}(p+q-2k+1)=(p+1)(q+1),
\] the dimension of all bidegree-\((p,q)\) polynomials. Hence these
images are a direct-sum decomposition of that space. Equation (P13),
with \(N=p+q-2k\), then determines every coefficient of the
decomposition and yields (P12). \(\square\)

For \(p=q=1\), formula (P12) decomposes \(x_1y_2\) into \[
\tfrac12(x_1y_2+x_2y_1)+\tfrac12 [xy].
\] For \(F=f(x)g(y)\) of orders \(p,q\), the first lesson's transvectant
normalization gives \[
h_k=\frac{p!q!}{(p-k)!(q-k)!}(f,g)_k.
\] Substitution into (P12) fixes the constants of the usual transvectant
series.

For any number of binary pairs, repeatedly apply (P12) to two pairs,
treating the other pairs as coefficient variables. Their diagonal form
introduces one pair \(z\); expand that pair with the next one and
continue. At every stage (P11) and the displayed factorial coefficient
specify the polar and contraction explicitly. The number of stages and
the number of terms at each stage are finite. This gives a complete
expansion for several binary series of variables, rather than only an
existence assertion.

\subsection{8. A general polarized series with finite matrix
coefficients}\label{section-8}

The decomposition into a few-variable polynomial and polarization
operators also has an explicit finite form in arbitrary dimension. The
following construction gives coefficients by rational matrix operations;
it does not require a classification of irreducible representations.

On \(H_d\), use the Fischer inner product over \(\mathbb C\): \[
\langle x^\alpha,x^\beta\rangle=\delta_{\alpha\beta}\alpha!.
\] Multiplication by a variable is adjoint to its derivative, so
\(D_{ij}^*=D_{ji}\). A \textbf{primary polynomial} will mean one killed
by all \(D_{ij}\), \(i<j\). By Lemma 3.2 every primary polynomial uses
at most the first \(n\) rows.

\protect\phantomsection\label{theorem-8-1}{}

\textbf{Theorem 8.1 (finite general series).} For fixed \(n,m,d\), there
are explicit rational operators \(R_W\), all built from the \(D_{ij}\),
such that \protect\phantomsection\label{equation-P14}{}

\[
F=\sum_W Wf_W,\qquad f_W=R_WF,
\tag{P14}
\] where every \(f_W\) is primary and \(W\) is a word in polarization
operators. If \(F\) is invariant under any diagonal group, every \(f_W\)
is invariant. The construction is valid over every characteristic-zero
field.

\textbf{Proof and construction.} In the monomial basis of \(H_d\), form
the rational matrix \[
Q=\sum_{i<j}D_{ji}D_{ij}.
\] It is positive semidefinite, since
\(\langle QF,F\rangle=\sum_{i<j}\|D_{ij}F\|^2\). Its kernel is exactly
the primary subspace. Write its characteristic polynomial as
\(t^s h(t)\) with \(h(0)\ne0\). The spectral theorem makes \[
P=h(Q)/h(0)
\] the orthogonal projection onto that kernel: it is one on the zero
eigenspace and zero on every nonzero eigenspace. If \(Q=0\), take
\(P=I\). This describes \(P\) by rational matrix entries.

Let \(a=\dim H_d\), and take all words \(W\) of length at most \(a\) in
the \(D_{ij}\), including the empty word. Their images of
\(\operatorname{im}P\) span \(H_d\). Indeed, successive word spans form
an ascending chain; once a step stops increasing, the span is stable
under every \(D_{ij}\). It then contains the orbit span of every primary
vector. Complete reducibility and Lemma 3.2 make that orbit span the
whole space. Before stabilization there can be at most \(a\) strict
increases.

Form \[
T=\sum_W WPW^*.
\] It is positive definite: if \(\langle TF,F\rangle=0\), then
\(PW^*F=0\) for every \(W\), making \(F\) orthogonal to all
\(W\operatorname{im}P\), which span the space. Thus \(T\) is invertible.
Compute \(T^{-1}\) by its adjugate divided by its nonzero determinant,
and set \[
R_W=PW^*T^{-1}.
\] Now \(\sum_WWR_W=TT^{-1}=I\), proving (P14), and
\(R_WF\in\operatorname{im}P\). Every operator used commutes with a
diagonal group by Proposition 1.1; inverses of invertible commuting
operators also commute. This proves the invariance assertion.

All entries and identities are rational. Their verification over
\(\mathbb C\) therefore proves the same identities over every
characteristic-zero field. The derivative argument in Lemma 3.2 proves
that the primary images there also use only the first \(n\) rows.
\(\square\)

This is a matrix form of the general series principle behind the
Capelli--Mertens--Deruyts reduction: a polynomial is a finite sum of
polars of primary polynomials, with invariant operations extracting the
primary coefficients. It is less economical computationally than the
specialized classical formulas, but gives a full finite construction and
exact coefficients for any number of series. Theorem 5.1 applies it
separately to each repeated kind of coefficient vector. The binary
formula (P12) is much shorter because its primary pieces can be indexed
by a single contraction number.

\subsubsection{Determinant powers and their
generalization}\label{determinant-powers-and-their-generalization}

There is a more structured version for \(r\leq n\) rows of \(n\)
variables. For a coordinate subset \(I\subset\{1,\ldots,n\}\),
\(|I|=r\), put \[
\Delta_I=\det(x_{ia})_{1\leq i\leq r,\ a\in I},
\qquad
\Omega_I=\det(\partial_{x_{ia}})_{1\leq i\leq r,\ a\in I},
\] in increasing coordinate order. A polynomial is
\textbf{determinant-harmonic} if every \(\Omega_I\) kills it.

\protect\phantomsection\label{theorem-8-2}{}

\textbf{Theorem 8.2 (general determinant series).} Every homogeneous
polynomial in \(r\leq n\) rows is a finite sum of products of their
\(r\)-row determinants times determinant-harmonic polynomials. Each
harmonic coefficient is a finite sum of polars of polynomials in only
\(r-1\) rows. All extraction coefficients can be computed by rational
finite matrices. For \(r=n\) the expansion has the unique form
\protect\phantomsection\label{equation-P17}{}

\[
F=\Phi_0+\Delta\Phi_1+\cdots+\Delta^\mu\Phi_\mu,
\qquad \Omega\Phi_i=0.
\tag{P17}
\] For \(r>n\), Theorem 8.1 first reduces to \(n\) rows; thus these
constructions give the series for every number of rows.

\textbf{Proof.} On each degree piece the adjoint of multiplication by
\(\Delta_I\), in the Fischer inner product, is \(\Omega_I\). Hence
\protect\phantomsection\label{equation-P18}{}

\[
H_d=\bigcap_I\ker\Omega_I\ \oplus\
\sum_I\Delta_I H_{d-r}
\tag{P18}
\] as an orthogonal direct sum of the indicated harmonic space and the
sum of determinant multiples. The second sum can have internal
relations; (P18) does not assert its summands are independent.

Here is a rational computation of the decomposition. The matrix \[
K_d=\sum_I \Delta_I\Omega_I:H_d\to H_d
\] is positive semidefinite, and its kernel is the harmonic space. If
its characteristic polynomial is \(t^s h(t)\), \(h(0)\ne0\), put
\(P_0=h(K_d)/h(0)\). As before, this is the harmonic projector; if
\(K_d=0\), take \(P_0=I\). Choose a rational polynomial \(q(t)\) such
that \(q(0)=0\) and \(tq(t)\equiv1\pmod{h(t)}\); the Euclidean algorithm
gives an inverse of \(t\) modulo \(h\), and adding a multiple of \(h\)
arranges \(q(0)=0\). Put \(K_d^+=q(K_d)\). On nonzero eigenspaces it is
the inverse of \(K_d\), and on its kernel it is zero. Therefore
\protect\phantomsection\label{equation-P19}{}

\[
F=P_0F+\sum_I\Delta_I F_I,\qquad
F_I=\Omega_I K_d^+F.
\tag{P19}
\] Every \(F_I\) has degree \(d-r\). Reapply (P19) to each of them.
Degree decreases at every step, so the procedure terminates in products
of minors times harmonic coefficients. It uses only rational matrices
and rational inverses.

It remains to justify the claimed polars of \(r-1\) rows. The exterior
proof of Capelli's identity works unchanged when the number \(n\) of
coordinate columns exceeds the number \(r\) of rows: extract
\(e_1\cdots e_r\) from (P7), retaining all coordinate subsets. It gives
the rectangular identity \protect\phantomsection\label{equation-P20}{}

\[
\operatorname{cdet}(D_{ij}+(r-j)\delta_{ij})_{1\leq i,j\leq r}
=\sum_{|I|=r}\Delta_I\Omega_I.
\tag{P20}
\] The harmonic space is stable under row substitutions. Indeed, row
multiplication transforms each differential determinant by its
determinant character, so its zero condition is preserved. Split it into
simple row modules by Lemma 3.1. In each summand choose a maximal row
weight \(F\), as in Lemma 3.2. Write that weight as
\((\lambda_1,\ldots,\lambda_r)\). In the column determinant on the left
of (P20), every nonidentity permutation term kills \(F\): take its
largest moved column \(j\). Its selected row is less than \(j\), so the
factor \(D_{\sigma(j),j}\) is a raising operator; all factors to its
right are diagonal and act by scalars. The identity term acts by \[
\prod_{j=1}^r(\lambda_j+r-j).
\] The right side kills a harmonic \(F\), so this product is zero. The
first \(r-1\) factors are positive integers; characteristic zero makes
them nonzero. Hence \(\lambda_r=0\), and \(F\) is independent of its
last row. Its orbit generates the summand. Thus every harmonic
polynomial is a sum of polars of polynomials on \(r-1\) rows.

The coefficient extraction can be made explicit by the \(P,W,T^{-1}\)
construction of Theorem 8.1 on the same degree space. All of its
operators are polynomials in the \(D_{ij}\), or inverses of invertible
such matrices, so they preserve the harmonic subspace. Its extracted
primary coefficients are therefore harmonic; the argument with (P20)
makes each independent of row \(r\). This supplies the promised finite
rational coefficients.

When \(r=n\), there is only one minor \(\Delta\). Multiplication by it
is injective in the polynomial ring. Equation (P18) then gives a unique
harmonic coefficient and a unique quotient, at every degree step,
proving uniqueness of (P17). The proof by Fischer positivity establishes
rational operator identities; those identities descend unchanged to any
characteristic-zero field. \(\square\)

Noether's \emph{On series expansion in form theory} (1920), the addendum
to Part III of her 1916 paper, explicitly corrects the earlier use of
\(\Omega(\Delta\psi)=c_1\psi+c_2\Delta\Omega\psi\) as a general
identity. It is valid for binary rows but not for arbitrary dimension.
Equations (P18)--(P20) supply the necessary general projection and the
quotient separately, so this proof does not make that assumption. In
dimension two, (P17) specializes to the explicitly normalized binary
series in Section 7.

\subsection{9. Symbols that stand for coefficients}\label{section-9}

The expression \(f=a_x^d\) in classical invariant theory is
\textbf{symbolic}. It does not assert that a general form is a power of
a linear form. Introduce a row of independent symbol variables
\(a=(a_1,\ldots,a_n)\), and write \(a_x=\sum a_rx_r\). The linear
\textbf{umbral evaluation} for this letter sends \[
a^\alpha\longmapsto A_\alpha\quad (|\alpha|=d),
\] where \(f=\sum_{|\alpha|=d}(d!/\alpha!)A_\alpha x^\alpha\). Then
\(U(a_x^d)=f\).

For several distinct letters of degree \(d\), apply this rule to each
letter and multiply their images. Thus, for binary forms,
\protect\phantomsection\label{equation-B22}{}

\[
U(a_1^{d-r}a_2^r b_1^{d-s}b_2^s)=A_rA_s.
\tag{B22}
\] For a system of forms, assign each letter to one of the forms and use
that form's degree and coefficients. The map is defined on the subspace
having exactly the assigned degree in each letter. It is linear, and it
respects products that use disjoint sets of letters. It is generally not
multiplicative on arbitrary symbolic polynomials involving the same
letter.

Use the determinants \[
[ab]=a_1b_2-a_2b_1,\qquad
[abc]=\det\begin{pmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{pmatrix}
\] in dimensions two and three, respectively. The \(n\)-dimensional
bracket has \(n\) covector rows.

\protect\phantomsection\label{theorem-9-1}{}

\textbf{Theorem 9.1 (equivariant evaluation).} Umbral evaluation
intertwines changes of coordinates. A product of brackets and factors
\(a_x\), having the assigned degree in each letter, evaluates to a
covariant. Its determinant weight is the number of brackets, counted
with multiplicity.

\textbf{Proof.} A change \(g\) acts on every letter by
\(a\mapsto ag^{-1}\). For one letter the identity \[
U_f((ag^{-1}x)^d)=f(g^{-1}x)
\] follows by expanding the power. Comparing the coefficients of
\(x^\alpha\) proves the intertwining identity on the basis of
degree-\(d\) symbol monomials. Tensoring these identities proves it for
several letters, because their evaluations multiply. Each bracket scales
by \(\det(g)^{-1}\), while \((ag^{-1})\cdot(gx)=a_x\). Apply the
intertwining identity to the bracket product. This gives
\href{NOE-INV-01.html\#equation-B1}{(B1)} with the stated weight.
\(\square\)

In particular the normalized Hessian has the symbolic expression
\protect\phantomsection\label{equation-B23}{}

\[
(f,f)_2=U\bigl([ab]^2a_x^{d-2}b_x^{d-2}\bigr).
\tag{B23}
\] For a direct proof, apply \(\Omega^2\) to \(a_x^db_X^d\). It gives
\(d^2(d-1)^2 [ab]^2a_x^{d-2}b_X^{d-2}\); the factorial factor in
\href{NOE-INV-01.html\#equation-B3}{(B3)} cancels that coefficient.
Evaluation and differentiation commute, and then \(X=x\) gives (B23).

Symbolic descriptions are not unique. For four binary letters, expansion
of the \(2\times2\) determinants gives
\protect\phantomsection\label{equation-B24}{}

\[
[ab][cd]-[ac][bd]+[ad][bc]=0.
\tag{B24}
\] For example, the coefficient of \(a_1b_2c_1d_2\) in the first two
terms is \(1-1\), and the same cancellation, after permuting the letters
and indices, cancels every monomial.

The converse to Theorem 9.1 is also useful: every polynomial covariant
is a linear combination of these evaluations. Its input is the mixed
first fundamental theorem, Theorem 6.2, proved in Section 6 above. To
see how the input applies, polarize a degree-\(\delta\) polynomial in
the coefficients into \(\delta\) separate coefficient slots. In each
slot replace a degree-\(d\) form by \(a_x^d\), using a new letter.
Powers span the space of forms: expanding \((t_1x_1+\cdots+t_nx_n)^d\)
and comparing the coefficients of the \(t_r\) proves this, since the
multinomial coefficients are nonzero. The polarized covariant therefore
becomes an invariant polynomial in covector letters and the vector
\(x\). The mixed first fundamental theorem writes that polynomial in
covector determinants and pairings \(a_x\); a determinant of vector
slots is zero when all those slots are the single repeated vector \(x\).
Keep the specified multidegree in the letters, then evaluate and
restitute the coefficient slots. This recovers the original covariant.
In dimension two, the same conclusion follows directly from the vector
theorem by identifying \(x\) with the covector \((-y,x)\).

The determinant identity (B24) is proved here. Theorem 4.2 above proves
that the Plücker shuffle relations generate the full relation ideal over
every commutative coefficient ring; Corollary 4.3 gives integral vector
invariants.

\subsection{10. Orthogonal vectors and quadratic ground
forms}\label{section-10}

\subsubsection{Orthogonal invariants in dimension
two}\label{orthogonal-invariants-in-dimension-two}

Over \(\mathbb C\), write a vector as \((x_i,y_i)\), and put
\(u_i=x_i+iy_i,\ v_i=x_i-iy_i\). The special orthogonal group acts by
\(u_i\mapsto tu_i,\ v_i\mapsto t^{-1}v_i\). Indeed these substitutions
preserve \(x_i^2+y_i^2=u_iv_i\), and every determinant-one isometry has
this form in the isotropic basis. A reflection interchanges \(u_i,v_i\),
and these two components give the whole orthogonal group.

Torus-invariant monomials have equal total \(u\)- and \(v\)-degrees, so
they are products of \(u_iv_j\). Put \[
q_{ij}=u_iv_j+v_iu_j,\qquad r_{ij}=u_iv_j-v_iu_j.
\] The torus ring is generated by the \(q_{ij},r_{ij}\), since \(2\) is
invertible. The reflection fixes the \(q\)'s and negates the \(r\)'s.
Averaging a polynomial expression with its reflected expression leaves
only monomials with an even number of \(r\)-factors. Expansion gives \[
r_{ij}r_{kl}=q_{il}q_{jk}-q_{ik}q_{jl}.
\] Pairing up those factors expresses every invariant in the \(q\)'s.
Since \(q_{ij}=2(x_ix_j+y_iy_j)\), the orthogonal invariant ring is
generated by the pairwise inner products.

\subsubsection{Binary quadratics retain
orientation}\label{binary-quadratics-retain-orientation}

For quadratics \[
f_i=a_ix^2+2b_ixy+c_iy^2
\] define \protect\phantomsection\label{equation-P15}{}

\[
h_i=a_ic_i-b_i^2,\quad
M_{ij}=a_ic_j+c_ia_j-2b_ib_j,\quad
\tau_{ijk}=\det\begin{pmatrix}
a_i&b_i&c_i\\a_j&b_j&c_j\\a_k&b_k&c_k
\end{pmatrix}.
\tag{P15}
\] The first two are the discriminants and their polarizations: the
coefficient of \(t\) in \(h(f_i+tf_j)\) is \(M_{ij}\). The determinant
is also special-linear invariant. In fact the determinant of the
coefficient representation \(\operatorname{Sym}^2(g^{-1})\) is
\(\det(g)^{-3}\). One can check it on diagonal matrices, where its three
eigenvalues multiply to that expression, and then on every matrix by
polynomial density.

\protect\phantomsection\label{theorem-10-1}{}

\textbf{Theorem 10.1 (joint quadratic invariants).} In characteristic
zero, the invariants of any number of binary quadratics are generated by
the \(h_i,M_{ij},\tau_{ijk}\). For two quadratics the three generators
\(h_1,h_2,M_{12}\) suffice.

\textbf{Proof.} Polarize any invariant in the coefficients into separate
quadratic slots. The symbolic completeness proved by Theorem 6.2 and
umbral evaluation expresses it as a sum of bracket monomials with degree
two in each symbol letter. Such a monomial is a graph: its vertices are
letters, its bracket factors are edges, and every vertex has degree two.
A loop is zero. Every remaining component is a cycle, with a doubled
edge allowed as a cycle of length two.

For a symbol column \(a\), let \[
J=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad
N_a=aa^{\mathsf t}J.
\] The product of brackets around a cycle is \[
[a_1a_2]\cdots [a_ra_1]
=\operatorname{tr}(N_{a_1}\cdots N_{a_r}).
\] This follows by multiplying the rank-one matrices: every intervening
scalar is \(a_i^{\mathsf t}Ja_{i+1}\), and the trace supplies the last
bracket. Umbral evaluation in a quadratic slot gives the traceless
matrix \[
N(f_i)=\begin{pmatrix}-b_i&a_i\\-c_i&b_i\end{pmatrix}.
\] Thus the cycle evaluates to the trace of a word in those matrices,
with a possible sign when an edge orientation is reversed.

For any traceless \(2\times2\) matrices \(A,B\), direct multiplication
gives \protect\phantomsection\label{equation-P16}{}

\[
AB+BA=\operatorname{tr}(AB)I.
\tag{P16}
\] We claim that all word traces are polynomials in traces of lengths
two and three. For a word of length \(r\geq4\), exchanging adjacent
letters using (P16) changes its trace to the negative trace of the
exchanged word plus a product of a length-two trace and a
length-\((r-2)\) trace. Repeating adjacent exchanges expresses its
trace, modulo sums of such shorter products, as an alternating
multilinear function of its \(r\) matrix arguments: antisymmetrize over
all \(r!\) permutations and divide by \(r!\). The alternating part is
zero, because the vector space of traceless \(2\times2\) matrices has
dimension three and an alternating \(r\)-linear function on that space
is zero for \(r>3\). This proves the claim by induction, with length-one
traces zero.

Multiplication gives \(\operatorname{tr}(N(f_i)N(f_j))=-M_{ij}\),
including \(M_{ii}=2h_i\). Equation (P16) makes the length-three trace
alternating, so it is a scalar multiple of the determinant of the three
coefficient rows. On the rows \((1,0,0),(0,1,0),(0,0,1)\), the trace is
\(-1\); hence it is \(-\tau_{ijk}\). Consequently every cycle evaluation
is in the algebra generated by (P15). Products of cycles and restitution
prove generation for the original invariant. With only two quadratic
kinds, every alternating triple has repeated rows and is zero, leaving
the three stated generators. \(\square\)

The triple determinant is necessary as soon as there are three
quadratics. The two ordered triples \[
(x^2,\,2xy,\,y^2),\qquad(y^2,\,2xy,\,x^2)
\] have identical self-discriminants and all pairwise \(M_{ij}\), but
their \(\tau_{123}\) are \(1\) and \(-1\). Thus a polynomial in the
pairwise data cannot equal \(\tau_{123}\). The coefficient reflection
\((a,b,c)\mapsto(c,b,a)\) preserves the bilinear discriminant form and
reverses orientation. Special-linear substitutions on the binary
variables preserve that orientation.

\subsection{Exercises}\label{exercises}

\protect\phantomsection\label{exercise-1}{}

\textbf{Exercise 1 --- The row Lie algebra.} Verify (P3) by applying
both sides to a coordinate function, then explain why this suffices.

\hyperref[solution-1]{Solution}.

\protect\phantomsection\label{exercise-2}{}

\textbf{Exercise 2 --- Capelli on determinant powers.} Check (P8) on
\(\Delta^s\), where \(\Delta=x_{11}x_{22}-x_{12}x_{21}\) and \(s\geq1\).

\hyperref[solution-2]{Solution}.

\protect\phantomsection\label{exercise-3}{}

\textbf{Exercise 3 --- Two quadratic kinds.} Prove that the invariants
of two binary quadratics are generated by their discriminants and
polarized discriminant.

\hyperref[solution-3]{Solution}.

\protect\phantomsection\label{exercise-4}{}

\textbf{Exercise 4 --- A direct binary vector proof.} Prove bracket
generation for binary vectors without Capelli, and organize the argument
as induction in the number of rows.

\hyperref[solution-4]{Solution}.

\protect\phantomsection\label{exercise-5}{}

\textbf{Exercise 5 --- More quadratic kinds and the missing generator.}
Determine whether Weyl's theorem reduces an arbitrary number of binary
quadratics to two quadratics. Give the correct bound and generating
patterns.

\hyperref[solution-5]{Solution}.

\protect\phantomsection\label{exercise-6}{}

\textbf{Exercise 6 --- Two quartic bracket expressions.} Evaluate
\([ab]^4\) and \([ab]^2 [ac]^2 [bc]^2\) for letters assigned to the same
quartic.

\hyperref[solution-6]{Solution}.

\subsection{Solutions}\label{solutions}

\subsubsection{Solution 1}\label{solution-1}

We have \(D_{ij}x_{ra}=\delta_{jr}x_{ia}\), so \[
[D_{ij},D_{kl}]x_{ra}
=\delta_{lr}\delta_{jk}x_{ia}
-\delta_{jr}\delta_{li}x_{ka}.
\] This equals \((\delta_{jk}D_{il}-\delta_{li}D_{kj})x_{ra}\). The
commutator of two derivations is a derivation, as expansion of the two
product rules shows. A derivation on a polynomial ring is determined by
its values on its coordinate generators, proving the identity.

\hyperref[exercise-1]{Exercise 1}.

\subsubsection{Solution 2}\label{solution-2}

Each row Euler operator gives \(D_{ii}\Delta^s=s\Delta^s\), and
\(D_{12}\Delta^s=0\): replacing row two by row one in the determinant
gives zero. Thus the left side is \(s(s+1)\Delta^s\). For the right
side, differentiate: \[
\partial_{11}\partial_{22}\Delta^s
=s\Delta^{s-1}+s(s-1)x_{11}x_{22}\Delta^{s-2},
\] \[
\partial_{12}\partial_{21}\Delta^s
=-s\Delta^{s-1}+s(s-1)x_{12}x_{21}\Delta^{s-2}.
\] Their difference is \(s(s+1)\Delta^{s-1}\). Multiplication by
\(\Delta\) agrees with the left side. For \(s=1\) the terms multiplied
by \(s(s-1)\) are zero and the derivative difference is \(2\). This
verifies the assigned test functions; Theorem 2.1 proves the operator
equality on every polynomial.

\hyperref[exercise-2]{Exercise 2}.

\subsubsection{Solution 3}\label{solution-3}

Theorem 10.1 expresses every invariant as a polynomial in
\(h_1,h_2,M_{12}\) and triple determinants. A triple drawn from two
kinds repeats a coefficient row, so its determinant is zero. The
remaining three are invariant by (P15), proving equality. They are also
algebraically independent: on \(f_1=2bxy\), their values are
\(-b^2,\ \alpha\gamma-\beta^2,\ -2b\beta\). When \(b\ne0\), these three
quantities can be prescribed freely on a dense open set over the
algebraic closure, by choosing \(b,\beta\) and then \(\alpha,\gamma\). A
polynomial relation therefore vanishes identically.

\hyperref[exercise-3]{Exercise 3}.

\subsubsection{Solution 4}\label{solution-4}

For one row, special-linear transitivity on nonzero vectors gives only
constants. For two rows, the determinant-orbit argument gives
\(k [[12]]\). For \(m>2\), apply the row-maximal-weight proof in Section
3 with \(n=2\): any simple row-substitution component of the invariant
space contains a vector killed by \(D_{ij}\), \(i<j\); the equations
\(D_{1j}F=D_{2j}F=0\) and invertibility of the first two rows force
independence of every later row. Its orbit generates that component from
a polynomial in \([12]\). A row translate of \([12]\) is a linear
combination of brackets \([ij]\), so each component, and then the entire
invariant space, is in their algebra. This proves the induction step for
every new row at once, using the two-row base case. It uses complete
reducibility and row derivatives, with no Capelli identity. The
determinant proof in Section 4 is its \(n\)-dimensional version.

\hyperref[exercise-4]{Exercise 4}.

\subsubsection{Solution 5}\label{solution-5}

The coefficient representation has dimension three, so the bound is
three. The triples displayed after Theorem 10.1 give a counterexample to
generation solely by self- and pairwise discriminants: all those values
agree, while \(\tau_{123}\) changes sign. Theorem 10.1 gives the
generating patterns are \(h_i,M_{ij},\tau_{ijk}\). These are precisely
the polarizations of the corresponding invariants on three copies. The
pair and triple generators have coefficient degrees two and three,
respectively. The copy bound is three, the dimension of the coefficient
representation.

\hyperref[exercise-5]{Exercise 5}.

\subsubsection{Solution 6}\label{solution-6}

Distinguish the symbol letters from the coefficient names \(a,b,c,d,e\).
The first expansion gives \[
U([ab]^4)=
\sum_{r=0}^4(-1)^r\binom4r A_rA_{4-r}
=2ae-8bd+6c^2=2i.
\] The second evaluation is an invariant of coefficient degree three.
Since the quartic invariant ring is \(k [i,j]\), it is a scalar multiple
of \(j\). To determine the scalar, use \(f=6x^2y^2\), so only \(A_2=1\)
is nonzero and \(j=-1\). The coefficient of
\(a_1^2a_2^2b_1^2b_2^2c_1^2c_2^2\) in the bracket product is \[
\sum_{r=0}^2(-1)^r\binom2r^3=1-8+1=-6.
\] To obtain this sum, select the second-index exponents \(r,2-r,r\) in
the three brackets \([ab]^2,[ac]^2,[bc]^2\); these are exactly the
choices giving degree two in each coordinate of each letter. Hence
\(U([ab]^2 [ac]^2 [bc]^2)=6j\).

\hyperref[exercise-6]{Exercise 6}.

\subsection{Further reading}\label{further-reading}

Noether's 1916 reduction paper and 1920 addendum treat repeated kinds
and determinant differential identities. Her1919 paper, Section 3,
treats integral binary brackets. Kraft and Procesi,
\href{https://dmi.unibas.ch/fileadmin/user_upload/dmi/Personen/Kraft_Hanspeter/Classical_Invariant_Theory.pdf}{Classical
Invariant Theory: A Primer}, Sections 7.1--7.4, provides another account
of polarization and vector invariants.

\end{document}