The Fischer Operator

Introduction

The Fischer decomposition writes the homogeneous polynomials of a hypercomplex system as layers of harmonic, or of monogenic, pieces multiplied by powers of a radial or a Clifford factor; this article treats the operator that performs the decomposition. The setting is the one of The Dirac Operator: a Clifford algebra $A=\mathrm{Cl}_{0,m}$ with frame $1,e_1,\dots,e_m$, a left Clifford module $\mathcal{S}$ of values, the vector variable $x$ ranging over $\mathbb{R}^{m+1}$, the Cauchy–Riemann operator $D=\sum_{\mu=0}^{m}e_\mu\partial_\mu$ and the Laplacian $\Delta=D\bar D=\bar DD$.

On the space $\mathcal{P}_k$ of $\mathcal{S}$-valued homogeneous polynomials of degree $k$ there are two decompositions. The harmonic one, classical and due to Stokes and Fischer, writes $\mathcal{P}_k=\bigoplus_j|x|^{2j}\mathcal{H}_{k-2j}$ with $\mathcal{H}_j=\ker\Delta\cap\mathcal{P}_j$ the harmonic polynomials; the monogenic one, its Clifford refinement, writes $\mathcal{P}_k=\bigoplus_j(x^{\natural})^j\mathcal{M}_{k-j}$ with $\mathcal{M}_j=\ker D\cap\mathcal{P}_j$ the solid spherical monogenics. Both are statements that a lowering operator — the Laplacian, or the Cauchy–Riemann operator — has a complementary raising operator — multiplication by $|x|^2$, or by the conjugate variable $x^{\natural}$ — whose images fill the polynomials degree by degree. The Fischer operator is the projection onto the harmonic, or the monogenic, part that this decomposition defines, and it is a polynomial expression in the raising and lowering operators.

The article establishes the two projections, the algebraic identities that make them computable, and the comparison between them: every monogenic polynomial is harmonic, so the monogenic part is a subspace of the harmonic part, while the two projections are different operators. The function-theoretic content of the decompositions — the Cauchy theory, the integral formulae, the spherical monogenics as such — is Clifford Analysis's and Regularity and the Cauchy–Riemann Operator's; the article reads the decompositions as operator statements and keeps the spherical-harmonic side, which is the same statement with the Laplacian in place of $D$, as the model.

The Euler Operator and the Grading

Definition. The Euler operator is

$$ E = \sum_{\mu=0}^{m}x_\mu\,\partial_\mu , $$

the infinitesimal generator of the dilations $x\mapsto\lambda x$; a polynomial $P$ is homogeneous of degree $k$ exactly when $EP=kP$, and $\mathcal{P}_k$ is the eigenspace of $E\subset\mathcal{P}$ with eigenvalue $k$.

Proposition (commutation with the grading). The Euler operator commutes with the Laplacian and with the Cauchy–Riemann operator, $[E,\Delta]=0$ and $[E,D]=0$, and it satisfies $[E,L_{|x|^2}]=2L_{|x|^2}$ and $[E,L_{x^{\natural}}]=L_{x^{\natural}}$, where $L_f$ denotes multiplication by $f$ on the left; consequently the lowering operators preserve the degree by $-2$ and $-1$ respectively and the raising operators increase it by $2$ and $1$.

Proof. The Laplacian and $D$ have constant coefficients, so $E$ commutes with them term by term because $x_\mu\partial_\mu$ and $\partial_\nu$ satisfy $[\partial_\nu,x_\mu\partial_\mu]=\partial_\nu$ and the second derivatives cancel. For the multipliers, $E(fP)=(Ef)P+f(EP)$ for a homogeneous $f$, giving $[E,L_f]=(\deg f)L_f$, and $\deg|x|^2=2$, $\deg x^{\natural}=1$. $\square$

Corollary (the grading of the two decompositions). The harmonic decomposition has layers of degree $k,k-2,k-4,\dots$ and the monogenic one has layers of degree $k,k-1,k-2,\dots$; the parity of the degree is preserved by the harmonic layers and not by the monogenic ones.

The Harmonic Fischer Operator

Lemma (Fischer, harmonic form). For $k\ge2$ the Laplacian $\Delta:\mathcal{P}_k\to\mathcal{P}_{k-2}$ is surjective, and multiplication by $|x|^2$ maps $\mathcal{P}_{k-2}$ injectively with image meeting $\mathcal{H}_k$ only at $0$.

Proof. The surjectivity follows from the map of the divisor: with the complexified symbol, $\Delta$ is induced by the quadratic form $\sum_\mu\xi_\mu^2$, which is nondegenerate, so the multiplication $\mathcal{P}_{k-2}\to\mathcal{P}_k$, $Q\mapsto|x|^2Q$, has kernel $0$ and the orthogonality is by the argument of Regularity and the Cauchy–Riemann Operator, where the lemma is stated and proved. $\square$

Theorem (the commutator identity). With $n=m+1$ the dimension of the ambient space,

$$ [\Delta, L_{|x|^2}] = \Delta L_{|x|^2}-L_{|x|^2}\Delta = 4E+2n . $$

Proof. Direct computation: $\partial_\mu(|x|^2Q)=2x_\mu Q+|x|^2\partial_\mu Q$, so $\partial_\mu^2(|x|^2Q)=2Q+4x_\mu\partial_\mu Q+|x|^2\partial_\mu^2Q$; summing over the $n$ coordinates gives $\Delta(|x|^2Q)=2nQ+4EQ+|x|^2\Delta Q$, which is the identity. $\square$

Corollary (the triangularity and the isomorphism). On $\mathcal{P}_{k-2}$ the composition $\Delta L_{|x|^2}$ is triangular with respect to degree, of the form $(4k+2m-6)\,\mathrm{id}+|x|^2\Delta$, so $\Delta:L_{|x|^2}\mathcal{P}_{k-2}\to\mathcal{P}_{k-2}$ is an isomorphism for $k\ge2$.

Proof. By the theorem, $\Delta(|x|^2Q)=(4E+2n)Q+|x|^2\Delta Q$; on $\mathcal{P}_{k-2}$ the first term is the scalar $4(k-2)+2n=4k+2m-6$ times $Q$, and the second raises the degree. The scalar does not vanish for $k\ge2$ and the degree-raising term is nilpotent on $\mathcal{P}_{k-2}$, so the map is invertible. $\square$

Definition (the harmonic Fischer operator). For $k\ge2$ the Fischer projection of degree $k$ is

$$ \pi^{\mathrm{har}}_k = I - L_{|x|^2}\circ\bigl(\Delta\big|_{L_{|x|^2}\mathcal{P}_{k-2}}\bigr)^{-1}\circ\Delta , $$

and for $k\le1$ it is the identity, $\mathcal{P}_k=\mathcal{H}_k$.

Theorem (the projection and the decomposition). The map $\pi^{\mathrm{har}}_k$ is a projection of $\mathcal{P}_k$ onto $\mathcal{H}_k$ with kernel $|x|^2\mathcal{P}_{k-2}$, and it is equivariant for the orthogonal group of the ambient space. Iterating gives

$$ \mathcal{P}_k = \mathcal{H}_k\oplus|x|^2\mathcal{H}_{k-2}\oplus|x|^4\mathcal{H}_{k-4}\oplus\cdots , $$

that is $\mathcal{P}_k=\bigoplus_{j\ge0}|x|^{2j}\mathcal{H}_{k-2j}$, the Stokes decomposition.

Proof. If $P=\pi^{\mathrm{har}}_kP$ then $\Delta P=0$ by construction, since $\Delta\pi^{\mathrm{har}}_kP =\Delta P-\Delta(|x|^2Q)$ with $Q=(\Delta|_{L_{|x|^2}\mathcal{P}_{k-2}})^{-1}\Delta P$; conversely a harmonic $P$ has $\Delta P=0$, so $Q=0$ and $\pi^{\mathrm{har}}_kP=P$. The map kills $|x|^2Q$ because $\Delta(|x|^2Q)$ is its own image under $\Delta$; idempotency follows. The equivariance is that of $\Delta$ and of $|x|^2$, both of which are rotation-invariant. Applying the projection repeatedly to the remainder gives the layered form. $\square$

Remark (the classical statement). The harmonic Fischer decomposition is the statement that the harmonic polynomials of all degrees, multiplied by the radial factors $|x|^{2j}$, exhaust the polynomials; it reduces the theory of spherical harmonics to the linear algebra of the spaces $\mathcal{H}_k$. It is Regularity and the Cauchy–Riemann Operator's, and for $m=0$, where there is no Clifford algebra and the operator is the radial derivative, it is the statement that the homogeneous polynomials of one variable are the powers $x_0^k$.

The Monogenic Fischer Operator

The same plan with the first-order operator $D$ in place of $\Delta$ and the Clifford multiplier $x^{\natural}$ in place of $|x|^2$ produces the Clifford refinement.

Lemma (Fischer, monogenic form; standard). For $k\ge1$ the operator $D:\mathcal{P}_k\to \mathcal{P}_{k-1}$ is surjective, and left multiplication by $x^{\natural}$ maps $\mathcal{P}_{k-1}$ injectively with image meeting $\mathcal{M}_k$ only at $0$.

Theorem (the monogenic Fischer decomposition). For every $k\ge0$,

$$ \mathcal{P}_k = \bigoplus_{j=0}^{k}\,(x^{\natural})^{\,j}\,\mathcal{M}_{k-j} , \qquad \dim\mathcal{M}_k = \dim\mathcal{S}\left[\binom{m+k}{k}-\binom{m+k-1}{k-1}\right] . $$

Proof. By the lemma, $\dim\mathcal{M}_k=\dim\mathcal{P}_k-\dim\mathcal{P}_{k-1}$ and $\mathcal{P}_k=\mathcal{M}_k\oplus x^{\natural}\mathcal{P}_{k-1}$; the first equality is the dimension count, the second the direct-sum statement, and iterating the second gives the layered form. The lemma and the resulting decomposition are quoted as standard, the proof being the triangular induction of Clifford Analysis. $\square$

Definition (the monogenic Fischer operator). Let $\pi^{\mathrm{mon}}_k$ be the projection of $\mathcal{P}_k$ onto $\mathcal{M}_k$ along $x^{\natural}\mathcal{P}_{k-1}$, which exists and is unique by the theorem. The Fischer operator of the Clifford theory is the direct sum

$$ \mathcal{F} = \bigoplus_{k\ge0}\pi^{\mathrm{mon}}_k : \mathcal{P}\longrightarrow\mathcal{M}, $$

defined on the space $\mathcal{P}$ of all $\mathcal{S}$-valued polynomials, mapping it onto the space $\mathcal{M}$ of all monogenic polynomials.

Theorem (characterisation). The operator $\mathcal{F}$ is a projection, $\mathcal{F}^2=\mathcal{F}$; its image is $\mathcal{M}$ and its kernel is $x^{\natural}\mathcal{P}$, the ideal of the polynomial algebra generated by the conjugate variable. It is equivariant for the orthogonal group and for the symmetry group of the system, and it commutes with the Euler operator.

Proof. A projection along a complementary subspace is idempotent by definition; the image is $\mathcal{M}$ and the kernel is $\bigoplus_k x^{\natural}\mathcal{P}_{k-1}=x^{\natural}\mathcal{P}$. The equivariance is that of $D$ and of the Clifford multiplication, which are both invariant under the orthogonal group, and commutation with $E$ is the grading statement of the first section. $\square$

Remark (the two projections compared). Every monogenic polynomial is harmonic, because $\Delta f=\bar DDf=0$ when $Df=0$, so $\mathcal{M}_k\subseteq\mathcal{H}_k$. The inclusion is proper for $m\ge2$; in the definition $\mathcal{M}_k=\ker D\cap\mathcal{P}_k$ against $\mathcal{H}_k=\ker\Delta\cap\mathcal{P}_k$, the operator $D$ is the stronger condition. Consequently $\mathcal{F}$ and the harmonic projection $\pi^{\mathrm{har}}$ are different operators with the same codomain behaviour only in degree $0$: $\mathcal{F}$ kills the ideal generated by $x^{\natural}$, while $\pi^{\mathrm{har}}$ kills the ideal generated by $|x|^2$, and the second ideal is contained in the first because $|x|^2=x^{\natural}x$. The example to keep in mind is $m=1$, where $D$ is the holomorphic operator, $\mathcal{M}_k$ is spanned by $z^k$, and $\mathcal{H}_k$ has complex dimension $2$ for $k\ge1$, so the inclusion $\mathcal{M}_k\subset\mathcal{H}_k$ is strict as soon as $k\ge1$.

Example (the complex case). For $m=1$ the monogenic decomposition is $\mathcal{P}_k=\bigoplus_{j=0}^{k}\bar z^{\,j}\mathbb{C}z^{k-j}$, the standard monomial basis of the homogeneous polynomials of degree $k$, and $\mathcal{F}$ is the projection onto the highest holomorphic part. The harmonic decomposition is $\mathcal{P}_k=\bigoplus_{j\ge0}|z|^{2j}\mathcal{H}_{k-2j}$, and the two are related by the identity $|z|^2=\bar zz$: iterating the harmonic layers and splitting each $\mathcal{H}_l$ into its monogenic tower reproduces the monogenic decomposition degree by degree. The comparison is the whole content of the refinement: the harmonic Fischer decomposition is the Laplacian's, the monogenic one is the Cauchy–Riemann operator's, and the second refines the first.

The Fischer Operator on a Hypercomplex System

Remark (the general form). The construction is not special to the Clifford case. Let $(A,D)$ be a hypercomplex system in the sense of Regularity and the Cauchy–Riemann Operator, with $D$ elliptic and $D\bar D=\Delta$; then the same three ingredients are available: the Euler operator $E$, the multiplier $|x|^2$ for the Laplacian (whose layer multiplier is the radial factor), and the multiplier dual to $D$ (the Clifford conjugate variable) for the first-order operator. The operator $\Delta$ is surjective on homogeneous polynomials and $\Delta:L_{|x|^2}\mathcal{P}_{k-2}\to\mathcal{P}_{k-2}$ is triangular with nonvanishing scalar part, so the harmonic Fischer projection exists for every admissible system; the monogenic projection exists whenever the Fischer lemma for $D$ does, which is the case under the standing hypotheses. The particular dimensions of the pieces are properties of the system, computed by the dimension recursion; the operator is the same.

Remark (why the multiplier is the conjugate variable). In the corpus's sign convention the operator $D$ carries $+e_i\partial_i$ and satisfies $D\bar D=\Delta$, so the dual multiplier — the one for which $D$ acts triangularly with positive diagonal — is $x^{\natural}=x_0-\sum_{i\ge1}x_ie_i$, whose Clifford norm is $|x|^2$; the conjugate operator $\bar D$ plays the mirror role and pairs with the plain variable $x$. The point is that the multiplier has the same Clifford sign as the operator, not the same coordinate sign, and this is what makes the pieces of the decomposition the standard monomials rather than a triangular transform of them. The corpus states the decomposition in the conjugate variable in Clifford Analysis; the article Regularity and the Cauchy–Riemann Operator states the same refinement with the plain variable, where the layers are still a decomposition but of the mirror pairing $(\bar D,x)$. A reader comparing the two must therefore read the multiplier together with the operator, and the two statements agree once the pairing is fixed.

Summary

On the space $\mathcal{P}_k$ of $\mathcal{S}$-valued homogeneous polynomials of degree $k$, the Euler operator $E=\sum_\mu x_\mu\partial_\mu$ is the grading, and the Laplacian and the Cauchy–Riemann operator commute with it. The harmonic Fischer lemma — $\Delta:\mathcal{P}_k\to\mathcal{P}_{k-2}$ is surjective and $|x|^2\mathcal{P}_{k-2}\cap\mathcal{H}_k=0$ — together with the commutator identity $[\Delta,L_{|x|^2}]=4E+2n$ makes $\Delta:L_{|x|^2}\mathcal{P}_{k-2}\to\mathcal{P}_{k-2}$ an isomorphism for $k\ge2$, and hence defines the harmonic Fischer operator $\pi^{\mathrm{har}}_k=I-L_{|x|^2}(\Delta|_{L_{|x|^2}\mathcal{P}_{k-2}})^{-1}\Delta$, a projection of $\mathcal{P}_k$ onto $\mathcal{H}_k$ with kernel $|x|^2\mathcal{P}_{k-2}$; iterating gives the Stokes decomposition $\mathcal{P}_k=\bigoplus_j|x|^{2j}\mathcal{H}_{k-2j}$. The monogenic refinement replaces $\Delta$ by $D$ and $|x|^2$ by the conjugate variable $x^{\natural}$: the monogenic Fischer decomposition is $\mathcal{P}_k=\bigoplus_{j=0}^{k}(x^{\natural})^j\mathcal{M}_{k-j}$, with $\dim\mathcal{M}_k=\dim\mathcal{S}[\binom{m+k}{k}-\binom{m+k-1}{k-1}]$, and the Fischer operator $\mathcal{F}=\bigoplus_k\pi^{\mathrm{mon}}_k$ is the projection onto the monogenic polynomials with kernel $x^{\natural}\mathcal{P}$. Since every monogenic polynomial is harmonic, $\mathcal{M}_k\subseteq\mathcal{H}_k$, strictly for $m\ge2$; the two projections differ and are related by $|x|^2=x^{\natural}x$. The construction is available on every admissible hypercomplex system, and the function-theoretic content of the decompositions is Clifford Analysis's and Regularity and the Cauchy–Riemann Operator's.

Summary of Notation

Symbol	Meaning
$\mathcal{P}_k$	$\mathcal{S}$-valued homogeneous polynomials of degree $k$
$E=\sum_\mu x_\mu\partial_\mu$	Euler operator; $EP=kP$ on $\mathcal{P}_k$
$\mathcal{H}_k=\ker\Delta\cap\mathcal{P}_k$	Harmonic homogeneous polynomials
$\mathcal{M}_k=\ker D\cap\mathcal{P}_k$	Solid spherical monogenics
$L_f$	Left multiplication by $f$
$[\Delta,L_{\|x\|^2}]=4E+2n$	The harmonic commutator identity, $n=m+1$
$\pi^{\mathrm{har}}_k$	Harmonic Fischer projection onto $\mathcal{H}_k$
$\mathcal{F}=\bigoplus_k\pi^{\mathrm{mon}}_k$	The Fischer operator, projection onto $\mathcal{M}$
$\ker\mathcal{F}=x^{\natural}\mathcal{P}$, $\operatorname{im}\mathcal{F}=\mathcal{M}$	Kernel and image of the Fischer operator
$\mathcal{P}_k=\bigoplus_j\|x\|^{2j}\mathcal{H}_{k-2j}$	Stokes decomposition
$\mathcal{P}_k=\bigoplus_{j}(x^{\natural})^j\mathcal{M}_{k-j}$	Monogenic Fischer decomposition