ARCHIVUM MATHEMATICUM (BRNO) Tomus 46 (2010), 355Ű375 FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS Fred Brackx∗ , Hennie De Schepper∗ , and Vladimír Souček‡ Abstract. Euclidean Cliford analysis is a higher dimensional function theory studying soŰcalled monogenic functions, i.e. null solutions of the rotation invariant, vector valued, Ąrst order Dirac operator ∂. In the more recent branch Hermitean Cliford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator ∂ J , leading to the system of equations ∂f = 0 = ∂ J f expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group U(n). In this paper we decompose the spaces of homogeneous monogenic polynomials into U(n)-irrucibles involving homogeneous Hermitean monogenic polynomials and we carry out a dimensional analysis of those spaces. Meanwhile an overview is given of so-called Fischer decompositions in Euclidean and Hermitean Cliford analysis. 1. Introduction In 1917 Ernst Fischer proved (see [18]) that, given a homogeneous polynomial q(X), X ∈ Rm , every homogeneous polynomial Pk (X) of degree k can be uniquely decomposed as Pk (X) = Qk (X) + q(X)R(X), where Qk (X) is a homogeneous polynomial of degree k satisfying q(D)Qk = 0, D being the diferential operator corresponding to X through Fourier identiĄcation (Xj ↔ ∂xj , j = 1, . . . , m) and R(X) is a homogeneous polynomial of suitable degree. If in particular q(X) = kXk2 , then q(D) is the Laplacian ∆m and Qk is harmonic, leading to the decomposition P(Rm ; C) = (1) ∞ M ∞ M r2p Hk (Rm ; C) k=0 p=0 m of the space P(R ; C) of complex valued polynomials into the spaces Hk (Rm ; C) of complex valued harmonic homogeneous polynomials of degree k. Cliford analysis (see e.g. [3, 14, 19, 21]), in its most basic form being a generalization to higher dimension of holomorphic function theory in the complex plane, ofers the possibility for a reĄnement of this decomposition (1). Indeed, denoting by m (e1 , . . . , em ) an orthonormal Pm basis of R , the polynomial q(X) may be chosen to be q(X) = X, where X = α=1 eα Xα is a real vector in the complex Cliford algebra Cm constructed over Rm ; the diferential operator q(D) then is the Dirac operator 2000 Mathematics Subject Classification: primary 30G35. Key words and phrases: Fischer decomposition, Cliford analysis. 356 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK Pm ∂ = α=1 eα ∂Xα and Qk is a k-homogeneous polynomial null solution of ∂, a soŰcalled spherical monogenic. This leads to the well-known Fischer decomposition in Euclidean Cliford analysis of the space P(Rm ; S) of homogeneous polynomials taking their values in an irreducible representation S of Cm . Such a representation S is called a spinor space and usually realized inside the Cliford algebra using a primitive idempotent (see Section 5). This Fischer decomposition reads: P(Rm ; S) = (2) ∞ M ∞ M X p Mk (Rm ; S) , k=0 p=0 m where Mk (R ; S) denotes the space of spinor valued monogenic homogeneous polynomials of degree k. In particular each harmonic k-homogeneous polynomial Hk , be it real, complex or spinor valued, may be split as (3) Hk = Mk + XMk−1 Mk and Mk−1 being monogenic homogeneous polynomials of the indicated degree. In the books [23, 12] and the series of papers [24, 10, 1, 2, 7, 15, 16, 17] so-called Hermitean Cliford analysis has emerged as a reĄnement of Euclidean Cliford analysis. Hermitean Cliford analysis is based on the introduction of an additional datum, a so-called complex structure J, intended to bring the notion of monogenicity closer to complex analysis. This complex structure induces an associated Dirac operator ∂ J , whence Hermitean Cliford analysis then focusses on the simultaneous null solutions of both operators ∂ and ∂ J , called Hermitean monogenic functions. The resulting function theory is still in full development, see [8, 25, 9, 6, 5]. It is clear that the traditional approach sketched above cannot be used to obtain a Fischer decomposition of harmonic homogeneous polynomials in terms of Hermitean monogenic homogeneous polynomials. However, a Hermitean monogenic Fischer decomposition was realized in [7] by means of a representation theoretical approach which will be explained further on. This implies however that it is possible to split any monogenic homogeneous polynomial in terms of homogeneous Hermitean monogenic ones, which was established in [13]. The aim of the underlying paper is threefold: (i) to give an alternative proof of the latter splitting, revealing the match between the monogenic and the Hermitean monogenic decompositions of a given harmonic polynomial; (ii) to use the Fischer decomposition formulae for a dimensional analysis of spaces of monogenic and Hermitean monogenic homogeneous polynomials, and meanwhile (iii) to give an overview of all Fischer decompositions in Euclidean and Hermitean Cliford analysis. 2. Clifford algebra: the basics Consider a real vector space E of dimension m, equipped with a real symmetric and positive deĄnite bilinear form B(X, Y ), X, Y ∈ E, with associated quadratic form Q(X) = B(X, X). The orthogonal group O(E) and the special orthogonal group SO(E) are deĄned as the groups of automorphisms, respectively orientation preserving automorphisms g, leaving the bilinear form B invariant: B(gX, gY ) = B(X, Y ) , ∀ X, Y ∈ E . FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 357 Now, let (e1 , . . . , em ) be a basis of E, which we assume to be orthonormal w.r.t. the bilinear form B, i.e. B(ej , ek ) = δjk , j, k = 1, . . . , m. The introduction of this basis leads to the identiĄcation O(E) ≃ O(m), through representation by (m × m)-matrices g = [gjk ], naturally satisfying the condition gg T = g T g = 1m with 1m the unit matrix of order m, while in the case of SO(E) ≃ SO(m), the additional condition det(g) = 1 holds. Turning to the complexiĄcation EC of E and the complexiĄcation BC of B, let us now consider the Cliford algebras Cℓ(E, −Q) over E and Cℓ(EC , −QC ) over EC . When identifying E with Rm these Cliford algebras are often denoted Rm and Cm respectively. The Cliford or geometric product is associative but nonŰcommutative. With respect to the chosen basis, it is governed by the rules e2α = −B(eα , eα ) = −1 , α = 1, . . . , m, eα eβ + eβ eα = 0 , α 6= β = 1, . . . , m . In standard Euclidean Cliford analysis, each vector X ∈ E with P components m (X1 , . . . , Xm ) ∈ Rm , is identiĄed with the real Cliford vector X = α=1 Xα eα . Its PmFischer dual is the Ąrst order Cliford vector valued diferential operator ∂ = α=1 eα ∂Xα , called the Dirac operator, which may also be obtained in a co-ordinate free way as a generalized gradient, see e.g. [1, 2]. It is precisely this Dirac operator which underlies the notion of monogenicity, a notion which is the higher dimensional counterpart of holomorphy in the complex plane. A smooth function f , deĄned on E or on EC and taking values in either the real or the complex Cliford algebra, is called left monogenic if and only if it fulĄlls the Dirac equation ∂[f ] = 0. The groups O(E) and SO(E) are doubly covered by the so-called pin group Pin(E) and spin group Spin(E) of the Cliford algebra, respectively, realized inside Cm as Pin(E) = {s ∈ Cℓ(E, −Q) : ∃k ∈ N, s = ω 1 . . . ω k , ω i ∈ S m−1 , i = 1, . . . , k} Spin(E) = {s ∈ Cℓ(E, −Q) : ∃k ∈ N, s = ω 1 . . . ω 2k , ω i ∈ S m−1 , i = 1, . . . , 2k} where S m−1 is the unit sphere in E; through coŰordinatization it holds that Pin(E) ≃ Pin(m) and Spin(E) ≃ Spin(m). Taking g ∈ SO(E), with corresponding pin element sg ∈ Spin(E), the action of g on a vector in E translates to Cliford language as X ′ = g[X] ←→ X ′ = sg Xs−1 g . Considering its induced action on a function F , which is given for a Cliford algebra valued function by the so-called H-representation H(s)[F (X)] = sF (s−1 Xs)s−1 and for a spinor valued function F by the so-called L-representation L(s)[F (X)] = sF (s−1 Xs) one has the commutation relations [∂, H(s)] = 0 and [∂, L(s)] = 0, whence it follows that the Dirac operator is invariant under this action, and so is the notion of monogenicity. A similar observation applies to Pin(E). We now introduce the building blocks of the Hermitean Cliford setting. To this end, we endow the space (E, B) with a so-called complex structure by choosing an 358 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK SO(E) element J for which J 2 = −1, creating in this way the Hermitean space (E, B, J). Clearly (det J)2 = (−1)m , forcing the dimension m of E to be even: in the Hermitean context we thus have to put m = 2n. In (EC , BC ) the projection operators 21 (1 ± iJ) create two isotropic subspaces o n 1 W ± = Z ± ∈ EC : Z ± = (1 ± iJ)X, X ∈ E 2 which constitute the direct sum decomposition EC = W + ⊕ W − . Extending the action of g ∈ SO(E) to vectors in EC by Z ± ∈ W ± 7→ g[Z ± ] = 12 (g[X] ± ig[JX]), the subspaces W ± will remain invariant if and only if g commutes with the complex structure J, or in other words, if g belongs to  SOJ (E) = g ∈ SO(E) : gJ = Jg . Similarly OJ (E) ⊂ O(E) is deĄned. Note that the orthonormal basis (e1 , . . . , e2n ) of E may always be chosen in such a way that the complex structure J is represented by the matrix   0 1n . J= −1n 0 For an arbitrary OJ (E) element the commutation relation with J then is reĆected in the speciĄc form of the corresponding matrix:   A B G= −B A with AAT + BB T = 1n and AB T − BAT = 0, implying that A ± iB both belong to the unitary group U(n). In other words:  OJ (2n) = G ∈ O(2n) : GJ = JG is isomorphic with U(n), and so is OJ (E). By means of the projection operators 12 (1 ± iJ), the basis (e1 , . . . , e2n ) gives rise to an alternative basis for EC , called the Witt basis: 1 1 fj = (1 + iJ)[ej ] = (ej − i en+j ) , j = 1, . . . , n 2 2 1 1 f†j = − (1 − iJ)[ej ] = − (ej + ien+j ) , j = 1, . . . , n . 2 2 It splits into separate bases (f1 , . . . , fn ) and (f†1 , . . . , f†n ) for W + and W − , respectively. The Witt basis elements satisfy the Grassmann relations fj fk + fk fj = 0 , f†j f†k + f†k f†j = 0 , j, k = 1, . . . , n including their isotropy: f2j = 0 = f†j 2 , j = 1, . . . , n, and the duality relations fj f†k + f†k fj = δjk , j, k = 1, . . . , n . (f†1 , . . . , f†n ) thus generates a Grassmann algebra, CΛ†n . The †Űnotation corresponds to a Hermitean Each of the sets (f1 , . . . , fn ) and respectively denoted by CΛn and conjugation in Cℓ(EC , −QC ), deĄned as follows: take µ ∈ Cℓ(EC , −QC ) arbitrarily, FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 359 i.e. µ = a + ib, with a, b ∈ Cℓ(E, −Q). Then µ† = a − ib where a and b are the traditional Cliford conjugates of a and b in Cℓ(E, −Q). The components of the real vector X are now denoted as (x1 , . . . , xn , y1 , . . . , yn ), and the corresponding Cliford vector X may thus be rewritten in terms of the Witt basis as n n X X X= (zj fj − zjc f†j ) (xj ej + yj en+j ) = j=1 j=1 where we have introduced the complex variables zj = xj + iyj and their complex conjugates zjc , j = 1, . . . , n. For vectors in the isotropic subspaces W ± of EC a similar identiĄcation results into n X 1 Z + = (1 + iJ)X ←→ z = zj f j 2 j=1 Z− = n X 1 zjc f†j (1 − iJ)X ←→ −z † = − 2 j=1 whence the relation X = Z + + Z − may be rewritten in Cliford language as X = z − z † . Similarly we arrive at the deĄnition of the Hermitean Dirac operators ∂z = n X f†j ∂zj and ∂ z† = n X fj ∂zjc = ∂ †z j=1 j=1 which are the Fischer duals of z and z † , and may be seen as reĄnements of the Euclidean Dirac operator since ∂ = 2(∂ †z − ∂ z ). As a side remark, observe that the above operators may also be obtained in another way, making explicit use of the complex structure J. Indeed, let X| = J(X) = n X J(ej )xj + J(en+j )yj = n X (ej yj − en+j xj ) j=1 j=1 then there arises a second, associated (or "twisted") Dirac operator ∂ J = J(∂) = 2n X J(eα )∂α = α=1 n X (ej ∂yj − en+j ∂xj ) j=1 corresponding to X|. We then have that 1 1 i (1 + iJ)[∂] = ∂ + ∂J 2 2 2 1 1 i 2∂ z = − (1 − iJ)[∂] = − ∂ + ∂ J . 2 2 2 A smooth function F taking its values in the complex Cliford algebra or in spinor space S is called Hermitean monogenic (or h-monogenic for short) if it is a simultaneous null solution of both Euclidean Dirac operators, i.e. if it fulĄlls the system ∂[F ] = 0 = ∂ J [F ] 2∂ †z = 360 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK or, equivalently, if it is a simultaneous null solution of both Hermitean Dirac operators, i.e. if it fulĄlls the system ∂ z [F ] = 0 = ∂ †z [F ] . Also the two Hermitean Dirac operators ∂ z and ∂ †z may be generated (as was the case for the Euclidean Dirac operator ∂) as generalized gradients, see [26] through projection on the appropriate invariant subspaces, which moreover guarantees the invariance of the considered system under the group action of OJ (2n) ≃ U(n), see [1, 2]. For further use, observe that the Hermitean vector variables and Dirac operators are isotropic on account of the properties of the Witt basis elements, i.e. (z)2 = (z † )2 = 0 and (∂ z )2 = (∂ †z )2 = 0 whence the Laplacian ∆ = −∂ 2 allows for the decomposition and factorization ∆ = 4(∂ z ∂ †z + ∂ †z ∂ z ) = 4(∂ †z + ∂ z )2 = −4(∂ †z − ∂ z )2 while also −(z − z † )2 = (z + z † )2 = z z † + z † z = |z|2 = |z † |2 = kXk2 = r2 . 3. Harmonic analysis We start with the space P(Rm ; C) of complex valued polynomials deĄned on R , considered as a module over the full orthogonal group O(m). The action of the group O(m) on polynomials in P(Rm ; C) is the regular representation: m [g · P ](X) = P (g −1 · X), g ∈ O(m), P ∈ P(Rm ; C), X ∈ Rm . Denoting by Hk (Rm ; C) the space of complex valued harmonic kŰhomogeneous polynomials, each of the spaces r2p Hk (Rm ; C) , p ∈ N0 := N ∪ {0} , k ∈ N0 m is a subspace of P(R ; C) which is invariant under the O(m) action. In addition, they form the constituents of the Fischer decomposition (1) of P(Rm ; C): (4) P(Rm ; C) = ∞ M ∞ M r2p Hk (Rm ; C) . k=0 p=0 m In (4), all O(m)-modules Hk (R ; C) are irreducible and mutually inequivalent. In particular the space Pk (Rm ; C) of k-homogeneous polynomials decomposes as k m Pk (R ; C) = ⌊2⌋ M r2p Hk−2p (Rm ; C) . p=0 Next we consider the space P(R2n ; C) of complex valued polynomials deĄned on Euclidean space of even dimension, however considered as an OJ (2n) ∼ = U(n)-module. The action of OJ (2n) on polynomials in P(R2n ; C) is given by [u · P ](X) = P (u−1 · X), u ∈ OJ (2n), P ∈ P(R2n ; C), X ∈ R2n . FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 361 Since each complex valued polynomial in (x1 , . . . , xn , y1 , . . . , yn ) may be written also as a polynomial in the variables (z1 , . . . , zn , z1c , . . . , znc ), i.e. P (X) = P (x1 , . . . , xn , y1 , . . . , yn ) = Pe(z1 , . . . , zn , z1c , . . . , znc ) we have to determine the polynomials Pe which are invariant under the action of U(n). As is wellŰknown the space of U(n)-invariant polynomials in P(R2n ; End(C))
2 4 2p is the space with basis 1, r , r , . . . , r , . . . where r2 = n X x2j + yj2 = n X zj zjc = |zj |2 j=1 j=1 j=1 n X The operator corresponding to the generator r2 is the Laplacian ∆= n X ∂x2j xj + ∂y2j yj = 4 n X ∂zj ∂zjc j=1 j=1 whence we are lead to consider the space of harmonic polynomials in the complex variables (z1 , . . . , zn , z1c , . . . , znc ). Its subspace HkC of harmonic kŰhomogeneous polynomials may be decomposed as HkC = k M Ha,k−a (R2n ; C) a=0 2n where Ha,b (R ; C) is the space of harmonic polynomials of bidegree (a, b), i.e. aŰhomogeneous in the variables zj and bŰhomogeneous in the variables zjc , i.e. Ha,b (λz1 , . . . , λzn , µz1c , . . . , µznc ) = λa µb Ha,b (z1 , . . . , zn , z1c , . . . , znc ) . This leads to the Fischer decomposition (5) P(R2n ; C) = ∞ M ∞ M k M r2p Ha,k−a (R2n ; C) k=0 p=0 a=0 2p where the constituents r Ha,k−a (R2n ; C), p ∈ N0 , k ∈ N0 , a = 0, . . . , k, are irreducible invariant subspaces under the action of U(n). In particular the space Pk (R2n ; C) of k-homogeneous polynomials decomposes as k m Pk (R ; C) = ⌊ 2 ⌋ k−2p M M p=0 r2p Ha,k−2p−a (Rm ; C) . a=0 Comparing the Fischer decompositions (4) and (5), it is clear that changing the symmetry group from O(2n) to its subgroup OJ (2n) ≃ U(n) results in considering the polynomials as functions of the complex variables (z1 , . . . , zn , z1c , . . . , znc ) and splitting the spaces of harmonic homogeneous polynomials according to bidegrees of homogeneity: k M Ha,k−a (R2n ; C) . Hk (R2n ; C) = a=0 362 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK 4. Euclidean Clifford analysis As mentioned in the introduction, the Fischer decompositions in terms of spherical harmonics may be reĄned by considering the spherical monogenics of Cliford analysis. To that end we consider the space P(Rm ; S) of spinor valued polynomials and the LŰaction of the group Pin(m) on it, given by [L(s) · P ](X) = s P (b s −1 Xs) , P ∈ P(Rm ; S) , s ∈ Pin(m), X ∈ Rm . We also need the action of Pin(m) on the space P(Rm ; End(S)), where End(S) is isomorphic as a vector space with the complex Cliford algebra Cm when m is even, or with its even part when m is odd. Let s 7→ sb denote the main involution on Cm for which ej 7→ −ej ; it has eigenvalues ±1, the corresponding eigenspaces being the even and odd part of the Cliford algebra. The action of Pin(m) on P(Rm ; End(S)) then is s −1 Xs)] sb −1 , [s · f ](X) = s f (b f ∈ P(Rm ; End(S)), s ∈ Pin(m), X ∈ Rm . The space of Pin(m)-invariant polynomials inside P(Rm ; End(S)) has the basis (1, X, X 2 , X 3 , . . . , X p , . . .), generating a unital superalgebra or Z2 -graded algebra spanC (1, X 2 , X 4 , . . .) ⊕ spanC (X, X 3 , X 5 , . . .) which reĆects the natural grading of the Cliford algebra by its decomposition into the even subalgebra and the odd subspace. The Pin(m)-invariant diferential operator corresponding, under natural duality, with the generator X of this graded algebra, is the Dirac operator ∂. Its polynomial null solutions are called spherical monogenics; we denote by Mk (Rm ; S) the space of spinor valued k-homogeneous spherical monogenics. Then each of the spaces X p Mk (Rm ; S), p ∈ N0 , k ∈ N0 , is an irreducible invariant subspace of P(Rm ; S) under the action of Pin(m), leading to the Fischer decomposition (2) of P(Rm ; S): (6) P(Rm ; S) = ∞ M ∞ M X p Mk (Rm ; S) . k=0 p=0 In (6), all Pin(m)-modules Mk (Rm ; S) are irreducible and mutually inequivalent. To see (6) as a reĄnement of (4) just take into account the Fischer decomposition of spherical harmonics in terms of spherical monogenics (see also (3)) (7) Hk (Rm ; S) = Hk (Rm ; C) ⊗ S = Mk (Rm ; S) ⊕ XMk−1 (Rm ; S) meaning that inside Hk (Rm ; S) an isomorphic copy of both Pin(m)-irreducible modules Mk (Rm ; S) and Mk−1 (Rm ; S) is realized by the trivial embedding and by the embedding factor X respectively; explicitly for Hk ∈ Hk (Rm ; S) one has  Hk = 1 +  X∂ X∂ Hk − Hk . m + 2k − 2 m + 2k − 2 FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 363 5. Hermitean Clifford analysis In this section we further explore the space P(R2n ; S) of S valued polynomials on Euclidean space of even dimension R2n . Here, we further decompose S as n M S(v) S= v=0 into its so-called homogeneous parts S(v) , v = 0, . . . , n, i.e. eigenspaces with Pn eigenvalue v for the left multiplication operator β c = j=1 fj f†j (see [11]). We want to obtain a decomposition of the space P(R2n ; S) into irreducible subspaces under the action of the group PinJ (2n), which is a double cover of OJ (2n) inside the Cliford algebra; this group can be deĄned as PinJ (2n) = {s ∈ Pin(2n) : ssJ = sJ s} √ where sJ = s1 s2 . . . sn , with sj = 22 (1 − ej en+j ), j = 1, . . . , n, is a Spin(2n) element corresponding to the complex structure J ∈ SO(2n) under the double covering of SO(2n) by Spin(2n). The action of PinJ (2n) on P(R2n ; S) is given by s −1 zs, sb −1 z † s), s · fe(z, z † ) = sfe(b fe ∈ P(R2n ; S) , s ∈ PinJ (2n) whereas its action on P(R2n ; End(S)) = P(R2n ; C2n ) is given by s −1 zs, sb −1 z † s)b s −1 s · fe(z, z † ) = sfe(b Observe the use of the Hermitean vector variables z and z † , which are PinJ (2n)-invariant elements in P(R2n ; End(S)). In fact it may be proven by invariance theory (see e.g. [20]) that the space of all PinJ (2n)-invariant polynomials is spanned by all possible words in z and z † :
spanC 1, z, z † , z z † , z † z, z z † z, z † z z † , z z † z z † , z † z z † z, . . .
(i) = spanC wl (z, z † ) : l = 0, 1, 2, . . . , i = 1, 2 with (1) w2r+1 (z, z † ) = |z|2r z (2) w2r+1 (z, z † ) = |z|2r z † w2r (z, z † ) = (z z † )r = |z|2r−2 z z † w2r (z, z † ) = (z † z)r = |z|2r−2 z † z (1) (1) (2) (2) and w0 = w0 = 1. This space becomes a unital graded superalgebra, inheriting its grading from the Z2 -grading on Cm . As a Ąrst step towards the decomposition aimed at, we will split P(R2n ; S) according to bidegree of homogeneity and to the homogeneous parts of spinor space: ∞ n M M (v) P(R2n ; S) = Pa,b a,b=0 v=0 (v) Pa,b 2n (v) with = Pa,b (R ; C) ⊗ S . Under the natural duality the generators z and z † of the above superalgebra correspond to the Hermitean Dirac operators ∂ z and ∂ †z . (v) So we consider the spaces Ma,b of S(v) valued Hermitean monogenic homogeneous 364 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK polynomials of bidegree (a, b) in the variables (z1 , . . . , zn , z1c , . . . , znc ), the latter denoted as (z, z † ). This leads to the Fischer decomposition of the space of spinor valued polynomials according to the action of PinJ (2n) (see [7]): P(R2n ; S) = ∞ M n  M (v) Ma,b ⊕ ∞ M M p=1 i=1,2 a,b=0 v=0  (v) wp(i) (z, z † ) Ma,b . (v) In particular, for the space Ha,b of S(v) valued harmonic homogeneous polynomials of bidegree (a, b), this Fischer decomposition reduces to (v) (v) (v−1) (v+1) Ha,b = Ma,b ⊕ z Ma−1,b ⊕ z † Ma,b−1   zz † z†z (v) Ma−1,b−1 − ⊕ b−1+v a−1+n−v (8) (v) where we put Ma,b = {0} whenever a < 0, b < 0, v < 0 or v > n and moreover, (v) when b − 1 + v = 0 the last summand reduces to zz † Ma−1,b−1 , while, when (v) a − 1 + n − v = 0 it reduces to z † zMa−1,b−1 . Special attention should be paid to the cases where v = 0 and v = n. Indeed, for v = 0, Hermitean monogenicity means holomorphy (see [2]), so in this case the spaces of spherical Hermitean monogenics are simply the spaces of scalar valued holomorphic homogeneous polynomials in the variables (z1 , . . . , zn ), which implies that b must be zero. For v = n Hermitean monogenicity means anti-holomorphy, so in that case we end up with anti-holomorphic homogeneous polynomials in the variables (z1c , . . . , znc ), implying that a must be zero. This leads to the following special Fischer decompositions: for v = 0 one has (0) (0) • Ha,0 = Ma,0 ; (0) (1) (0) (1) (0) • Ha,1 = z † Ma,0 ⊕ z z † Ma−1,0 ; • Ha,b = z † Ma,b−1 , when b 6= 0, b 6= 1, while for v = n one has (n) (n) • H0,b = M0,b ; (n) (n−1) (n) (n−1) • H1,b = z M0,b (n) ⊕ z † z M0,b−1 ; • Ha,b = z Ma−1,b , when a 6= 0, a 6= 1. Note that the dimensional analysis carried out in Section 7 conĄrms these results. In the next section we will show how the decompositions (7) and (8) Ąt together, more precisely we will determine the U(n)-irreducible parts of (8) constituting each of the terms in (7). FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 365 6. Decomposition of Mk into U(n)-irreducibles Let us Ąrst decompose the space Pk (R2n ; S) of spinor valued k-homogeneous polynomials in the variables (z1 , . . . , zn , z1c , . . . , znc ) according to bidegree of homogeneity and to the homogeneous parts of spinor space: n M M (v) Pa,b Pk (R2n ; S) = a+b=k v=0 in this way inducing on a spherical monogenic Mk ∈ Mk (R2n ; S) the splitting (9) Mk = n k X X (v) Pa,k−a . a=0 v=0 (v) It is important to note that the components Pa,k−a are no longer monogenic since (v) (v−1) (v) (v+1) ∂ z : Pa,k−a −→ Pa−1,k−a ; ∂ †z : Pa,k−a −→ Pa,k−a−1 whence (v) (v−1) (v+1) ∂ : Pa,k−a −→ Pa−1,k−a ⊕ Pa,k−a−1 (v) with Pa,b = {0} whenever a < 0 or b < 0 or v < 0 or v > n. In other words: the action of the Dirac operator ∂ mixes up the homogeneous parts of spinor space. (v) (v) Introducing the spaces Ma,k−a = Mk (R2n ; S) ∩ Pa,k−a we clearly have that n k M M (v) Ma,k−a ⊂ Mk (R2n ; S) a=0 v=0 (v) (v) (v) (v) Moreover the polynomials in Ma,k−a satisfy ∂Ma,k−a = 0 = ∂ †z Ma,k−a − ∂ z Ma,k−a , (v) (v+1) (v) (v−1) where ∂ †z Ma,k−a ∈ Pa,k−a−1 and ∂ z Ma,k−a ∈ Pa−1,k−a . This means that at the (v) (v) (v) same time ∂ z Ma,k−a = 0 and ∂ †z Ma,k−a = 0, or: Ma,k−a is Hermitean monogenic, (v) which justiĄes the notation Ma,k−a for the corresponding space. We thus have (v) Lemma 1. On each of the spaces Pa,b the notions of monogenicity and Hermitean monogenicity coincide. Introducing the space of spherical Hermitean monogenics of degree k:  HMk = Qk ∈ Pk (R2n ; S) : ∂ z Qk = 0 = ∂ †z Qk we thus have obtained that k n M M (v) (10) Ma,k−a ⊂ HMk ⊂ Mk (R2n ; S) . a=0 v=0 However, there is more. Denoting the restrictions to Mk of the Hermitean Dirac f† f and ∂ operators by ∂ we have the following result. z z 366 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK Proposition 1. One has n k (v) f† M M f = Ker ∂ HMk = Ker ∂ Ma,k−a . z = z a=0 v=0 Proof. In view of (10) we still need to prove that n k M M (v) Ma,k−a ⊃ HMk . a=0 v=0 So take φk ∈ HMk , then φk = n k X X (v) φa,k−a a=0 v=0 (v) with φa,k−a (v) ∂ z φa,k−a = ∈ (v) Pa,k−a . As (v) ∂ z φa,k−a (v−1) ∈ Pa−1,k−a it follows from ∂ z φk = 0 that 0 for a = 1, . . . , k and v = 1, . . . , n, while for a = 0 or v = 0 this (v) equation is trivially satisĄed. Similarly it follows from ∂ †z φk = 0 that ∂ †z φa,k−a = 0 for a = 0, . . . , k − 1 and v = 0, . . . , n − 1, which now is trivial for a = k or v = n. (v) (v) We may thus conclude that φa,k−a ∈ Ma,k−a for a = 0, . . . , k and v = 0, . . . , n, which proves the statement.  Remark 1. Both for the spherical harmonics and for the spherical Hermitean monogenics, the decomposition according to bidegree of homogeneity and to spinor homogeneity leads to harmonic, respectively Hermitean monogenic components. For spherical monogenics however this is not the case, as already mentioned, since the action of the Dirac operator, in fact a combined action of both Hermitean Dirac operators, mixes up the homogeneous spinor subspaces. We can only say that the corresponding components of a spherical monogenic are in Ker ∂ z ∂ †z = Ker ∂ †z ∂ z . Our aim now is to decompose Mk into irreducible subspaces which are invariant under PinJ (2n) ∼ = U(n). To that end we start from the orthogonal decomposition f†
⊥ f ⊕ Ker ∂ Mk = Ker ∂ z z for which we have already shown in Proposition 1 that n k (v) f† M M f = Ker ∂ Ma,k−a . = Ker ∂ z z a=0 v=0 f f. f whence it suices to determine Im ∂ We also know that (Ker ∂ †z )⊥ ∼ = Im ∂ z z Lemma 2. One has (11) f⊂ Im ∂ z k−1 n−1 M M a=0 v=1 (v) Ma,k−a−1 . FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 367 Proof. Using (9) and invoking that ∂ z Mk = ∂ †z Mk since Mk is monogenic we have k−1 X X n−1 (v+1) ∂ z Pa+1,k−a−1 = a=0 v=1 and hence k−1 X X n−1 (v−1) ∂ †z Pa,k−a a=0 v=1 (1) ∂ z Pa+1,k−a−1 = 0 ; (n−1) ∂ †z Pa,k−a = 0 ; (v+1) (v−1) (v) ∂ z Pa+1,k−a−1 = ∂ †z Pa,k−a ∈ Ma,k−a−1 from which the desired result follows.  To show that equality holds in (11) we prove the following version of the Poincaré Lemma. (v) (v) Lemma 3. Given φa,k−a−1 ∈ Ma,k−a−1 , the polynomial   z z† (v) ψ= φ + a + n − v k − a − 1 + v a,k−a−1 enjoys the following properties: (i) ψ ∈ Mk (v) (ii) ∂ z ψ = ∂ †z ψ = φa,k−a−1 Proof. To prove (ii) it suices invoke the well-known antiŰcommutation relations (see [7]): {∂ z , z} = Ez + n − β c ; {∂ z , z † } = 0 ; {∂ †z , z} = 0 ; {∂ †z , z † } = Ezc + β c (v) the Hermitean Euler operators Ez and Ez† having the spaces Pa,b as eigenspaces with respective eigenvalues a and b. Next, (i) follows from (ii).  Proposition 2. One has k−1 n−1 (v) f† M M f = Im ∂ Ma,k−a−1 . = Im ∂ z z a=0 v=1 Proof. In view of (11) we still have to prove that To that end take φ= f⊃ Im ∂ z k−1 X X n−1 a=0 v=1 (v) k−1 n−1 M M (v) Ma,k−a−1 . a=0 v=1 φa,k−a−1 ∈ k−1 n−1 M M a=0 v=1 (v) Ma,k−a−1 368 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK and deĄne the polynomial Ψ= k−1 X X n−1 a=0 v=1   z z† (v) φ . + a + n − v k − a − 1 + v a,k−a−1 Then Ψ will belong to Mk and satisfy ∂ z Ψ = ∂ †z Ψ = φ.  Combining the above results we obtain the following Fischer decomposition, which, as mentioned in the introduction, was already obtained in [13] on the basis of group representation theory. Theorem 1. The space Mk (R2n ; S) of spinor valued spherical monogenics of degree k may be decomposed into U(n)-irreducibles as follows: (12) Mk = n k M M (v) Ma,k−a a=0 v=0 ⊕  k−1 n−1 M M a=0 v=1   z z† (v) Ma,k−a−1 . + a+n−v k−a−1+v This last result means that, given a spinor valued spherical monogenic Mk , (9), (v) (v) there exist spinor valued spherical Hermitean monogenics fa,k−a and ga,k−a−1 such that Mk = n k X X (v) fa,k−a + (v−1) n k X X z ga−1,k−a a=1 v=2 a=0 v=0 a+n−v + k−1 X X n−2 (v+1) z † ga,k−a−1 a=0 v=0 k−a+v where the polynomials occuring in the respective projections from Mk onto the U(n)-irreducibles involving spherical Hermitean monogenics, may be calculated as (v) fa,k−a (v−1)  = 1− z∂ z a+n−v (v) ga−1,k−a = ∂ z Pa,k−a (v+1) − z † ∂ †z k−a+v  (v) Pa,k−a (v) ga,k−a−1 = ∂ †z Pa,k−a Now we are able to show explicitly how the Fischer decomposition (8) in terms of spherical Hermitean monogenics originates from the Fischer decomposition (7) in terms of standard spherical monogenics, by using the decomposition (12) of Theorem 1. First we have, according to (7): Hk (R2n ; S) = Mk (R2n ; S) ⊕ (z − z † )Mk−1 (R2n ; S) FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 369 which, by means of (12), takes the form Hk (R2n ; S) = k n M M (v) Ma,k−a a=0 v=0 k−1 n−1 M M  ⊕ a=0 v=1 ⊕ (z − z † )    z z† (v) Ma,k−a−1 + a+n−v k−a−1+v n k−1 M M (v) Ma,k−a−1 a=0 v=0 ⊕ (z − z † ) k−2 n−1 M M  a=0 v=1    z† z (v) + Ma,k−a−2 . a+n−v k−a−2+v This means that for each spherical harmonic Hk ∈ Hk (R2n ; S) there exist spherical Hermitean monogenics (v) (v) fa,k−a ∈ Ma,k−a (v) ga,k−a−1 (v) ha,k−a−1 (v) (a = 0, . . . , k; v = 0, . . . , n) ∈ (v) Ma,k−a−1 (a = 0, . . . , k − 1; v = 1, . . . , n − 1) ∈ (v) Ma,k−a−1 (a = 0, . . . , k − 1; v = 0, . . . , n) (v) ua,k−a−2 ∈ Ma,k−a−2 (a = 0, . . . , k − 2; v = 1, . . . , n − 1) such that Hk = n k X X a=0 v=0 +   k−1 X  X n−1 (v) fa,k−a + n  k−1 X X a=0 v=1 (v) (z − z † ) ha,k−a−1 a=0 v=0 +  k−2 X  X n−1 a=0 v=1    z z† (v) ga,k−a−1 + a+n−v k−a−1+v  zz † z † z  (v) ua,k−a−2 . − k−a−2+v a+n−v Fixing a bidegree and a spinor-homogeneity degree the above decomposition yields (v) meaning that (v)  g (v−1) a−1,k−a  g (v+1)   a,k−a−1 (v−1) (v+1) + ha−1,k−a + z † − ha,k−a−1 a+n−v k−a+v   zz † z†z (v) + u − k − a − 1 + v a − 1 + n − v a−1,k−a−1 (v) Ha,k−a = fa,k−a + z (v) (v−1) (v+1) Ha,k−a = Ma,k−a ⊕ z Ma−1,k−a ⊕ z † Ma,k−a−1   z†z zz † (v) Ma−1,k−a−1 − ⊕ k−a−1+v a−1+n−v 370 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK which is precisely (8). Moreover we may now also determine the projection operators (v) from Ha,k−a onto the U(n)-irreducibles involving spherical Hermitean monogenics. With the notations from above we successively obtain: 1 (v) (v) ua−1,k−a−1 = ∂ † ∧ ∂ z [Ha,k−a ] k+n−1 z     g (v+1) 1 z a,k−a−1 (v) (v+1) ∂ †z + − ha,k−a−1 = ∂ †z ∧ ∂ z [Ha,k−a ] k−a+v k−a+v a−1+n−v    g (v−1)   1 z† a−1,k−a (v−1) (v) + ha−1,k−a = ∂ †z ∧ ∂ z Ha,k−a ∂z − a+n−v a+n−v k−a−1+v and  z † ∂ †z  (v) z∂ z (v) [Ha,k−a ] − fa,k−a = 1 − a+n−v k−a+v   z † z(∂ †z ∧ ∂ z ) zz † (∂ †z ∧ ∂ z )  (v) − Ha,k−a − (k − a + v)(k + n − 1) (a + n − v)(k + n − 1) 7. Dimensional analysis Fischer decompositions of spaces of polynomials allow for dimension counting, which we will do in a systematic way in this section, Ąrst conĄrming well-known formulae for the spaces of spherical harmonics and spherical monogenics, and then establishing a dimension result for spaces of spherical Hermitean monogenics. First recall that   m+k−1 k dim(Pk (Rm ; C)) = Dm = k k Dm denoting the number of k-combinations of an m-element set, repetition being k allowed. It follows that dim(Pk (R2n ; S)) = 2n D2n , since dim(S) = 2n . In the same order of ideas we have   n (v) a b (v) Dna Dnb dim(Pa,b ) = Dn Dn dim(S ) = v and observe that indeed n X X (v) dim(Pa,b ) = a+b=k v=0 ! k n   X  X
n k = dim Pk (R2n ; S) . Dna Dnk−a = 2n D2n v a=0 v=0 Next, from the Fischer decomposition (1) it follows that Pk (R2n ; S) = Hk (R2n ; S) ⊕ r2 Pk−2 (R2n ; S) which yields for k ≥ 2


hk ≡ dim Hk (R2n ; S) = dim Pk (R2n ; S) − dim Pk−2 (R2n ; S)   2n + 2k − 2 2n + k − 2 k−2 k = 2n (D2n − D2n ) = 2n 2n + k − 2 k FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 371 while h0 = 2n , h1 = 2n+1 n. In the same order of ideas we Ąnd for a > 0 and b > 0  
n (v) (v) 2n (Dna Dnb − Dna−1 Dnb−1 ) ha,b ≡ dim Ha,b (R ; S) = v     n n+a−2 n+b−1 n+a+b−1 = n+b−1 v a b yielding n k X X a=0 v=0 k X
(v) (Dna Dnk−a − Dna−1 Dnk−a−1 ) dim Ha,k−a (R2n ; S) = 2n n =2 a=0 k (D2n
k−2 − D2n ) = dim Hk (R2n ; S) as it should. Now, from the Fischer decomposition (2) it follows that Pk (R2n ; S) = Mk (R2n ; S) ⊕ X Pk−1 (R2n ; S) yielding for k > 0
mk ≡ dim Mk (R2n ; S) = dim Pk (R2n ; S) − dim Pk−1 (R2n ; S) n =2 k (D2n − k−1 D2n ) n =2 k D2n−1 n =2   2n + k − 2 k while m0 = h0 = 2n . Note that k−2 k dim(Mk ) + dim(Mk−1 ) = 2n (D2n − D2n ) = dim(Hk ) which is in accordance with (7). (v) (v) (v) Finally, putting ma,b = dim(Ma,b ), with ma,b = 0 whenever a < 0 or b < 0 or v < 0 or v > n, we deduce from the Fischer decomposition (8) that (13) (v) (v) (v−1) (v+1) (v) ha,k−a = ma,k−a + ma−1,k−a + ma,k−a−1 + ma−1,k−a−1 (v) This means that the dimension of the spaces Ma,b of spherical Hermitean monogenics may be calculated recursively from the dimensions of the spaces of spherical harmonics: (v) (v) (v) (v) (v+1) (v) (v) (v−1) (v) (v) (v+1) (v) (v) (v+1) m0,0 = h0,0 m0,1 = h0,1 − h0,0 m1,0 = h1,0 − h0,0 m0,2 = h0,2 − h0,1 m1,1 = h1,1 − h1,0 (v+2) + h0,0 (v−1) − h0,1 (v) + h0,0 372 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK (v) (v) (v−1) (v) (v) (v+1) (v) (v) (v+1) (v) (v) (v−1) (v) (v) (v−1) m2,0 = h2,0 − h1,0 m0,3 = h0,3 − h0,2 m1,2 = h1,2 − h1,1 m2,1 = h2,1 − h1,1 m3,0 = h3,0 − h2,0 (v−2) + h0,0 (v+2) + h0,1 (v−1) − h0,2 (v+1) − h2,0 (v−2) + h1,0 (v+3) − h0,0 (v+2) + h1,0 (v) (v) (v+1) + h0,1 − h0,0 (v−2) + h1,0 + h0,1 (v−1) − h0,0 (v−3) − h0,0 etc. According to (12), these dimensions should satisfy mk = n k X X a=0 v=0 (v) ma,k−a + k−1 X X n−1 (v) ma,k−a−1 a=0 v=1 by means of which the correctness of the obtained results may be checked. However, solving the recurrence relations (13) explicitly in order to obtain a (v) closed form for ma,k−a turns out to be too complicated. Fortunately, the dimension of the spaces of spherical Hermitean monogenics may also be calculated in an alternative way. To this end we consider the Weyl dimension formula (see [20, p.301]) for the dimension of an irreducible Ąnite dimensional representation of a simple Lie algebra g. This formula contains products over all positive roots of g, the number of which is increasing quickly, whence the formula is diicult to use in explicit calculations. Yet, in some cases signiĄcant simpliĄcations occur. In particular in the present case, for representations of the algebra su(n), a simpliĄed formula may be used involving the soŰcalled hook numbers, see [20, p.382]. Characterizing an irreducible representation by its highest weight λ, a Young (or Ferrers) diagram may be associated to it, which consists of left justiĄed rows of boxes, each row containing as many boxes as indicated by the corresponding component of λ. Each box then has a hook number associated to its position in the diagram, which can be calculated following a simple rule: if there are x boxes in the diagram to the right of the considered one and y boxes below, then the hook number is x + y + 1. The Weyl dimension formula for the module with highest weight λ = [λ1 , λ2 , . . . , λn−1 ] then takes the form (λ1 + n − 1)! (λ2 + n − 2)! (λn−1 + 1)! 1 ··· (n − 1)! (n − 2)! 1! Πi,j∈λ hi,j where the product is taken over all hook numbers hi,j associated to all boxes in the diagram. (v) Now, the space Ma,b is a su(n)-module with highest weight λ = [a + b + 1, b + 1, . . . , b + 1, b, . . . , b] where the last b + 1 appears at the (n − v)-th place, see [13]. The corresponding Young diagram, with the hook numbers written in the corresponding boxes, is (v) shown above, leading for 0 < v < n to the following expression for dimMa,b :     b+n−1 a+n−1 a+b+n b+v−1 (v) ma,b = a+n−v b n−v−1 a FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 373 a+b +n − 1 ... a+n+1 a+n a+n−v b+n−2 ... n n−1 n−v−1 . . . . . . . . . . . . . . . b+v+1 ... v+3 v+2 2 b+v ... v+2 v+1 1 b+v−2 ... v v−1 . . . . . . . . . . . . b+1 ... 3 2 b ... 2 1 a ... 2 1 (v) Fig. 1: Ferrer diagram with hook numbers for Ma,b which has been checked to be in accordance with the recurrence relations (13). As already mentioned above for v = 0 the spaces of spherical Hermitean monogenics are nothing else but the spaces of scalar valued holomorphic homogeneous polynomials in the variables (z1 , . . . , zn ), implying that b = 0. Hence
(0) (0) ma,0 = dim Pa,0 = Dna = a+n−1 , which is conĄrmed by the Weyl dimension a formula for the highest weight [a, 0, . . . , 0]. Similarly, for v = n we end up with anti-holomorphic homogeneous polynomials in the variables (z1c , . . . , znc ), implying
(n) (n) that a = 0. Hence m0,b = dim P0,b = Dnb = b+n−1 , which is conĄrmed by the b Weyl dimension formula for the highest weight [b, b, . . . , b]. Acknowledgement. This research results from a joint project of the Cliford Research Group at Ghent University and the Mathematical Institute of Charles University Prague, supported by the bilateral scientiĄc co-operation between 374 F. BRACKX, H. DE SCHEPPER AND V. SOUČEK Flanders and Czech Republic. V. Souček acknowledges support by the institutional grant MSM 0021620839 and by grant GA CR 201/08/0397. References [1] Brackx, F., Bureš, J., Schepper, H. De, Eelbode, D., Sommen, F., Souček, V., Fundaments of Hermitean Clifford analysis – Part I: Complex structure, Compl. Anal. Oper. Theory 1 (3) (2007), 341Ű365. [2] Brackx, F., Bureš, J., Schepper, H. De, Eelbode, D., Sommen, F., Souček, V., Fundaments of Hermitean Clifford analysis – Part II: Splitting of h–monogenic equations, Complex Var. Elliptic Equ. 52 (10Ű11) (2007), 1063Ű1079. [3] Brackx, F., Delanghe, R., Sommen, F., Clifford Analysis, Pitman Publishers, 1982. [4] Brackx, F., Delanghe, R., Sommen, F., Differential forms and/or multi–vector functions, Cubo 7 (2) (2005), 139Ű169. [5] Brackx, F., Knock, B. De, Schepper, H. De, A matrix Hilbert transform in Hermitean Clifford analysis, J. Math. Anal. Appl. 344 (2) (2008), 1068Ű1078. [6] Brackx, F., Knock, B. De, Schepper, H. De, Sommen, F., On Cauchy and Martinelli–Bochner integral formulae in Hermitean Clifford analysis, Bull. Braz. Math. Soc. (N.S.) 40 (3) (2009), 395Ű416. [7] Brackx, F., Schepper, H. De, Eelbode, D., Souček, V., The Howe dual pair in Hermitean Clifford analysis, Rev. Mat. Iberoamericana 26 (2) (2010), 449Ű479. [8] Brackx, F., Schepper, H. De, Schepper, N. De, Sommen, F., Hermitean Clifford–Hermite polynomials, Adv. Appl. Cliford Algebras 17 (3) (2007), 311Ű330. [9] Brackx, F., Schepper, H. De, Sommen, F., A theoretical framework for wavelet analysis in a Hermitean Clifford setting, Commun. Pure Appl. Anal. 6 (3) (2007), 549Ű567. [10] Brackx, F., Schepper, H. De, Sommen, F., The Hermitian Clifford analysis toolbox, Adv. Appl. Cliford Algebras 18 (3Ű4) (2008), 451Ű487. [11] Brackx, F., Schepper, H. De, Souček, V., On the structure of complex Clifford algebra, submitted. [12] Colombo, F., Sabadini, I., Sommen, F., Struppa, D. C., Analysis of Dirac systems and computational algebra, Birkhäuser, Boston, 2004. [13] Damiano, A., Eelbode, D., Invariant operators between spaces of h–monogenic polynomials, Adv. Appl. Cliford Algebras 19 (2) (2009), 237Ű251. [14] Delanghe, R., Sommen, F., Souček, V., Clifford algebra and spinor-valued functions – A function theory for the Dirac operator, Kluwer Academic Publishers, Dordrecht, 1992. [15] Eelbode, D., Stirling numbers and Spin–Euler polynomials, Experiment. Math. 16 (1) (2007), 55Ű66. [16] Eelbode, D., Irreducible sl(m)–modules of Hermitean monogenics, Complex Var. Elliptic Equ. 53 (10) (2008), 975Ű987. [17] Eelbode, D., He, F. L., Taylor series in Hermitean Clifford analysis, Compl. Anal. Oper. Theory, DOI10.1007/s11785-009-0036-y. [18] Fischer, E., Über die Differentiationsprozesse der Algebra, J. für Math. 148 (1917), 1Ű78. [19] Gilbert, J., Murray, M., Clifford Algebra and Dirac Operators in Harmonic Analysis, Cambridge University Press, 1991. [20] Goodman, R., Wallach, N. R., Representations and Invariants of the Classical Groups, Cambridge University Press, 2003. [21] Gürlebeck, K., Sprössig, W., Quaternionic and Clifford Calculus for Physicists and Engineers, J. Wiley & Sons, Chichester, 1997. FISCHER DECOMPOSITIONS IN EUCLIDEAN AND HERMITEAN CLIFFORD ANALYSIS 375 [22] Lávička, R. Delangheand R., Souček, V., The Fischer decomposition for Hodge–de Rham Systems in Euclidean space, to appear. [23] Rocha-Chavez, R., Shapiro, M., Sommen, F., Integral theorems for functions and differential forms in Cm , vol. 428, Research Notes in Math., 2002. [24] Sabadini, I., Sommen, F., Hermitian Clifford analysis and resolutions, Math. Methods Appl. Sci. 25 (16Ű18) (2002), 1395Ű1414. [25] Sommen, F., Peija, D. Peija, A Martinelli–Bochner formula for the Hermitian Dirac equation, Math. Methods Appl. Sci. 30 (9) (2007), 1049Ű1055. [26] Stein, E. M., Weiss, G., Generalization of the Cauchy–Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163Ű196. ∗ Clifford Research Group, Faculty of Engineering, Ghent University, Galglaan 2, B-9000 Gent, Belgium ‡ Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic View publication stats