Multi-Centered Invariants, Plethysm and Grassmannians

Motivated by multi-centered black hole solutions of Maxwell-Einstein theories of (super)gravity in D=4 space-time dimensions, we develop some general methods, that can be used to determine all homogeneous invariant polynomials on the irreducible (SL_h(p,R) x G4)-representation (p,R), where p denotes the number of centers, and SL_h(p,R) is the"horizontal"symmetry of the system, acting upon the indices labelling the centers. The black hole electric and magnetic charges sit in the symplectic representation R of the generalized electric-magnetic (U-)duality group G4. We start with an algebraic approach based on classical invariant theory, using Schur polynomials and the Cauchy formula. Then, we perform a geometric analysis, involving Grassmannians, Pluecker coordinates, and exploiting Bott's Theorem. We focus on non-degenerate groups G4"of type E7"relevant for (super)gravities whose (vector multiplets') scalar manifold is a symmetric space. In the triality-symmetric stu model of N=2 supergravity, we explicitly construct a basis for the 10 linearly independent degree-12 invariant polynomials of 3-centered black holes.


Introduction
The Attractor Mechanism [1,2], originally discovered in N = 2, D = 4 Maxwell-Einstein supergravity and then investigated in other extended supergravities as well as in non-supersymmetric theories of gravity (see e.g. [3] for reviews and list of Refs.), plays a central role in the physics of extremal black holes (BHs), as well as of (intersecting configurations of) extremal black p-branes [4], also in D > 4 space-time dimensions. In its simplest framework, namely in presence of Abelian vectors and scalar fields in the background of an extremal BH, the area of the event horizon can be expressed purely in terms of the fluxes of the 2-form Abelian field strengths and of their duals, whose fluxes define the magnetic and electric BH charges, fitting a symplectic vector Q. The dynamics of the scalar fields exhibits an attractor phenomenon, namely the value of the field at the BH event horizon is completely determined in terms of the magnetic and electric charges, regardless of the initial (boundary) conditions defined for the flow at spatial infinity 1 . In general, the near-horizon attractor dynamics can be reformulated in terms of critical points of a BH effective potential [2], which in presence of an underlying local supersymmetry also enjoys a geometric interpretation in terms of central charge(s) and matter charges (if any).
The vector space of electric-magnetic BH charges generally defines an irreducible 2 representation (irrep.) space R for the generalized electric-magnetic (U -) duality 3 group G 4 . Under the action of G 4 , the irrep. space R undergoes a stratification into orbits, which in turn are in correspondence with classes of BHs, with both regular and vanishing near-horizon geometry (corresponding to "large" and "small" BHs, respectively); thus, the classification of G 4 -orbits in R results in a group-theoretical characterization of BH solutions themselves. In Maxwell-Einstein supergravity theories whose scalar manifold is a symmetric space G 4 /H 4 (with H 4 being the maximal compact subgroup of G 4 ), the classification of orbits can be algebraically achieved in terms of constraints imposed on the unique [9] algebraically independent G 4 -invariant homogeneous polynomial I in the irrep. R (see e.g. [10], as well as [11] for a recent résumé and a list of Refs.).
Within this rather broad class of D = 4 theories, I is a quadratic polynomial (I = I 2 ) for N = 2 minimally coupled [12,13] as well as for N = 3 [14] supergravity. In the remaining D = 4 theories with symmetric scalar manifolds, G 4 can be characterized (in terms of R) as a group "of type E 7 " [15,16,17,18,19,20]. In particular, the charge representation R satisfies dim ∧ 2 R = dim S 4 R = 1.
(1.1) Namely, the flux irrep. R is symplectic (i.e., endowed with a unique symplectic structure C [M N ] := 1 ∈ ∧ 2 R =: R ⊗2 a , as it generally holds in D = 4), and it exhibits a unique, algebraically independent, degree-4 homogeneous invariant polynomial 4 I = I 4 , related to a rank-4 completely symmetric G 4invariant tensor (the so-called K-tensor [21,22,23,24]) K (M N P Q) := 1 ∈ S 4 R =: R ⊗4 s . Simple and semi-simple non-degenerate U -duality groups G 4 "of type E 7 " relevant to the class of D = 4 Maxwell-Einstein (super)gravity theories under consideration are listed in Table 1 at the start of Sec.

2.3.
The properties of the quartic polynomial I 4 constructed from the K-tensor have been exploited in order to characterize in an algebraic way the various scalar flows in the background of extremal singlecentered BHs [10]. The classification can be extended to multi-centered BHs [25,26,27,28,29,30,31].
In the case of 2-centered solutions, a group theoretical study of the invariant structures which can be defined in the vector space of electric-magnetic fluxes has been started in [32], and then developed in [22,23,24,20]; the connection between 2-centered invariant structures for the so-called stu model [33] of N = 2, D = 4 supergravity and Quantum Information Theory has then been investigated in [34]. Furthermore, relations between the K-tensor of the stu model (giving rise to the so-called Cayley's hyperdeterminant [36,37]) and elliptic curves has been recently studied in [38], and extended to the 2-centered case in [34].
Besides the importance of the symplectic product W (see Eq. (2.17) below) in order to define mutually non-local charge vectors pertaining to different centers [25], the physical relevance of some higher-order U -invariant polynomials has been suggested in recent investigations [30], and further study in such a direction is surely deserved in order to unravel their role e.g. in the spatial structure of general stationary almost-BPS [27,29,31] and composite non-BPS [29,30,31] multi-centered BH flows, with flat D = 3 spatial slices as well as non-flat ones [28,39,40].
In the case of BH solutions with p centers, the U -duality group G 4 acts on p copies of R; correspondingly, the charge vectors Q a carry an index referring to the relevant center (a = 1, ..., p), and one has to consider polynomial invariants in the p dim R coordinates on R p . Thus, a "horizontal" symmetry 5 SL h (p, R), commuting with G 4 , naturally occurs. This was firstly introduced in [32], and it acts on the index labelling the various centers, in such a way that G 4 -invariant polynomials generally decompose into SL h (p, R)-irreps. In the 2-centered case (p = 2), as mentioned, the problem of determining a complete basis for the ring of (SL h (p, R) × G 4 )-invariant homogeneous polynomials has been solved in [32] and [22], respectively for semi-simple and simple non-degenerate groups "of type E 7 " occurring as U -duality groups in D = 4 supergravities with symmetric scalar manifolds 6 . Actually, the same results had been obtained, within a completely different approach based on nilpotent orbits, by Kac many years ago in [9]; therein, it was also shown that the complete basis composed by polynomials whose homogeneity degree is the lowest possible is also finitely generating, namely all other higher-order invariant polynomials are simply polynomials in the elements of the basis.
For example, in the 2-centered simple case [9,22] there are 7 algebraically independent U -invariant polynomials, which form a minimal degree complete basis for the corresponding ring; out of them, 5 are homogeneous of degree 4 and they are arranged into a 5 (spin s = 2) irrep. of the 2-centered "horizontal" symmetry SL h (2, R), while the remaining ones are polynomials homogeneous of degree 2 and 6 that are SL h (2, R)-invariant (the one of degree 2 is nothing but the symplectic product W defined in (2.17) below). Out of these 7 G 4 -invariants, one can construct 4 algebraically independent (SL h (2, R) × G 4 )-invariant polynomials, homogeneous of degree 2, 6, 8 and 12 [9,22]. With some abuse of language, (SL h (p, R) × G 4 )-invariants have been usually named "horizontal" invariants.
In the (2-centered) semi-simple case [9,32,24], further lower-order horizontal invariant structures arise as a consequence of the factorization of the U -duality symmetry G 4 ; a particular, noteworthy example is provided by the aforementioned stu model, exhibiting a triality symmetry [33], which should be modded out in order to obtain invariant structures relevant for BHs (cfr. the treatment of [32] vs. [34], as well as the treatment in Secs. 2.3.5 and 4.3.4).
Although some general properties can be inferred from elementary group theoretical considerations, a systematic study and classification of (p > 2)-centered solutions in terms of (SL h (p, R) × G 4 )-orbits is still lacking.
The aim of the present paper is to start developing some general methods that can be used to determine all invariants associated to p-centered BH solutions, for a generic p. In particular, we will be interested in p-centered horizontal invariants, namely homogeneous (SL h (p, R) × G 4 )-invariant polynomials on the irrep. R p ⊗R =: (p, R) of the overall symmetry SL h (p, R)×G 4 itself. The invariant polynomials homogeneous of degree k are clearly related to the (SL h (p, R) × G 4 )-invariant tensors in the k-th completely symmetric 7 power S k (R p ⊗ R) =: (p, R) ⊗k s . This allows for the exploitation of the classical invariant theory (for which we will mainly refer to the book [35]).
The plan of the paper is as follows. In Sec. 2 we use the representation theory of a product group G × G in order to determine the corresponding invariant structures. We first recall some general facts about invariant theory and, in particular, the characterization of the (G × G)-invariants in the symmetric products S k (U ⊗ V ) of the irreps. U and V of G and G, respectively. By applying these methods to the case G = SL h (p, R) and G = G 4 relevant to p-centered (BH) solutions in D = 4 supergravity, we can then count (SL h (p, R) × G 4 )-invariants 8 for all relevant generic, simple cases.
Next, in Sec. 3, we present a geometric analysis of the invariants. We show that in the p-centered case the invariants can be determined by using the Grassmannian Gr(p, R) of p-planes in R. This Grassmannian is embedded in a projective space by its Plücker coordinates, which are global sections of a line bundle L on Gr(p, R). For any positive integer a, the group GL(R), and thus 9 acts on the sections Γ(Gr(p, R), L ⊗a ). These sections are homogeneous polynomials of degree a in the Plücker coordinates. Our geometric characterization of the (SL h (p, R) × G 4 )-invariant polynomials, in combination with Bott's theorem [46], shows that all these invariants are given by (SL h (p, R) × G 4 )invariant sections. In particular, the (SL h (p, R) × G 4 )-invariant polynomials are generated by homogeneous polynomials in the Plücker coordinates. Finally, in Sec. 4, we present an application of the methods developed in Secs. 2 and 3 : in the semisimple, triality-symmetric N = 2, D = 4 stu model, we compute a basis for the 10-dimensional vector space of SL h (3, R) × SL(2, R) 3 -invariant polynomials homogeneous of degree 12 for 3-centered BHs; the physical issue of invariance under the symmetric group S 3 , implementing the triality symmetry acting on the three copies of SL(2, R) in the U -duality group G 4 = SL(2, R) 3 , is considered in Secs. 2.3.5 and 4.3.4.

Invariant Theory
In order to tackle the problem of determining the invariants associated to multi-centered BH solutions, we will make use of the classical invariant theory. Let us first collect some basic facts on how to find invariants in U ⊗ V for the action of the group GL(U ) × GL(V ); as mentioned above, we will mainly refer to the book [35], to which we address the reader for further details and a list of Refs.

The Schur Polynomials
A partition λ of an integer m ∈ Z >0 , denoted as λ ⊢ m, is a non-increasing sequence λ = (p 1 , . . . , p N ) of integers p i ∈ Z ≥0 such that N i= p i = m. The number of non-zero elements in λ is denoted by ht(λ) := n, so p i = 0 for i > n. 7 The subscript "s" ("a") stands for symmetric (antisymmetric) throughout. 8 Up to a certain order, fixed by the available computing power (see analysis in Sec. 2.3). 9 As also recently discussed in [42], the maximal (but generally non-symmetric) embedding G4 ⊂ Sp(R) (which in supergravity is named Gaillard-Zumino [43] embedding) can be regarded as a consequence of the following Theorem by Dynkin (Th. 1.5 of [44], more recently discussed e.g. in [45]) : every irreducible group of unimodular linear transformations of the N -dimensional complex space (namely, a group of transformations which does not leave invariant a proper subspace of such a space) is maximal either in SL(N ) (if the group does not have a bilinear invariant), or in Sp(N ) (if it has a skew-symmetric bilinear invariant), or in O(N ) (if it has a symmetric bilinear invariant). Exceptions to this rule are listed in Table VII of [45].
The Schur polynomial S λ in N variables x 1 , . . . , x N , where N ≥ n := ht(λ), is the symmetric polynomial, with integral coefficients, defined as the quotient ( [35] where the partition λ + ρ is defined as λ + ρ := (p 1 + N − 1, p 2 + N − 2, . . . , p N ), and A λ+ρ (x) and V (x) (Vandermonde determinant) are two anti-symmetric polynomials in x 1 , . . . , x N , respectively given by 2) where S N is the group of permutations of N variables and ǫ σ is the permutation parity, and which differ only in the possibility to consider or not the same values for at least a pair of indices in the string i 1 , . . . , i h .
For instance, λ := 1 h defines S λ (V ) := ∧ h V (2.4), the rank-h completely antisymmetric tensor representation of GL(V ), which has dimension N h ; in particular, the partition λ = 1 N selects the one-dimensional determinant representation on ∧ N V (realized by the Ricci-Levi-Civita symbol ǫ i 1 ...i N ). Another example is provided by the partition λ := k := (k, 0, . . . , 0 N −1 ), which defines S λ (V ) := S k V (2.5), the k-th symmetric product of V , namely the rank-k completely symmetric tensor representation of GL(V ). A basis of S k V is provided by v a 1 1 · · · v a N N with a i ≥ 0 and N i=1 a i = k, and the action of y on this basis elements is the multiplication by y a 1 1 · · · y a N N . Hence, the trace of y on S k V is the sum of all monomials in y 1 , . . . , y N which are homogeneous of degree k. As mentioned, this is the Schur polynomial S k (2.5), so tr(y|S k V ) = S k (y). A generating function for these S k can be obtained by noting that (1 + ... + y a 1 t a 1 + ...) (1 + ... + y a 2 t a 2 + ...) ... (1 + ... + y a N t a N + ...) and it is given by the Molien formula (S 0 (y) = 1; [35], 9.4.3, (4.4.3)): A generalization of the Molien formula (2.8), which yields the decomposition of S k (U ⊗ V ) under GL(U ) × GL(V ), is provided by the following formula, due to Cauchy. Let 10 m ≤ n be two positive integers, then: The interpretation of the Cauchy formula (2.9) in terms of characters of representations is given e.g. in [35], 9.6.3. Let U and V be vector spaces of dimension m and n respectively, and assume that m ≤ n. Let u 1 , . . . , u m , v 1 , . . . , v n be bases of U ,V respectively, and let x ∈ (C * ) m , y ∈ (C * ) n act on these spaces by diag(x 1 , . . . , x m ), diag(y 1 , . . . , y n ). The eigenvalues of (x, y) on U ⊗ V are then the x i y j with 1 ≤ i ≤ m and 1 ≤ j ≤ n. Thus, Cauchy formula (2.9) implies that (2.10) Using the bijection between traces of irreducible representations and irreducible characters, it follows that there is an isomorphism of (GL(U ) × GL(V ))-representations: A particular consequence of the isomorphism (2.11) is that if G × G is a subgroup of GL(U ) × GL(V ), then the vector space S k (U ⊗ V ) G×G of (G × G)-invariants in S k (U ⊗V ) enjoys the following decomposition: since the action of G×G on S λ (U )⊗S λ (V ) preserves the factors. Thus, in order to compute the (G × G)invariants, one can compute the G-invariants on all S λ (U ) and the G-invariants on all S λ (V ), and then combine the results. Given a partition λ = (p 1 , . . . , p N ), we define an integer k ∈ Z ≥0 and a partition µ with ht(µ) ≤ N − 1 by λ = (k, . . . , k) + (k 1 , . . . , k N −1 , 0) := (k N ) + µ. (2.13) Then, the restriction of S λ (V ) to SL(V ) is isomorphic to S µ (V ), since (k N ) is the k-th tensor product of the determinant representation. Ç a va sans dire, if ht(λ) > n, then the definition of S λ (V ) shows that it is the 0-dimensional vector space.

Application to p-centered Black Holes
As in Sec. 1, let G 4 be the U -duality group acting on the representation R in which the (fluxes of the) Abelian 2-form field strengths and their duals sit, in the background of a p-centered black hole solution in the corresponding D = 4 Maxwell-Einstein (super)gravity theory. Since the "horizontal" [32] group SL h (p) ≡ SL h (p, R) acts on the labels of the centers, in order to determine the invariants associated to the p-centered BH one has to compute the invariants of , is irreducible (if non-zero), and there are very few cases in which it is the trivial 1-dimensional representation. In fact, recall that S λ (V ) = 0 if ht(λ) > p, whereas if ht(λ) < p then S λ (V ) is an irreducible representation of GL(V ), and hence also of SL(V ). Thus S λ (V ) G = 0, unless S λ (V ) is a power of the 1-dimensional determinant representation of GL(V ); namely, unless the partition reads λ = (a, . . . , a) =: (a p ), in which case one has for some a ∈ Z ≥0 , and dim S λ (U ) SL h (U ) = 0 otherwise.
In virtue of formula (2.12), this implies that the invariants of SL h (p) × G 4 in S k (R p ⊗ R) must come from the invariants of G 4 in S λ (R) where λ = (a p ). As (a p ) ⊢ pa, it thus also follows 11 and there are no invariants 12  Before explicitly analyzing some cases relevant to supergravity, let us consider the lowest degrees of homogeneity : k = 2 and k = 3.

Homogeneity k = 2
In the case k = 2, the partitions λ with λ ⊢ 2 are λ = (2, 0) =: 2 and λ = (1, 1) =: (2.16) A particular case, in which the term S 2 U ⊗ S 2 V does not yield any invariant, is provided by 2-centered (p = 2) BHs in the framework under consideration, namely for p = 2: respectively (namely, both the fundamental spin s = 1/2 irrep. 2 of SL h (2, R) and the irrep. R of G 4 are symplectic) one obtains one invariant from the term ∧ 2 U ⊗ ∧ 2 V of (2.16), given by the symplectic product W in R of G 4 , namely [32,22,24] (a, b = 1, 2, M, N = 1, ..., dim R): where ǫ ab is the Ricci-Levi-Civita symbol of SL h (2, R). When W = 0, the charge vectors Q 1 and Q 2 (respectively pertaining to BH centers 1 and 2) are mutually non-local, and the distance between the two centers in the BPS 2-centered system is fixed [25]. No other algebraically independent invariant polynomial homogeneous of degree k = 2 arise, since the representations U = R 2 =: 2 of SL h (2, R) and V = R of G 4 are irreducible, and thus there are no other invariants in S 2 U and in S 2 V . As discussed at the end of Sec. 3 of [22], some SL h (p, R)-covariant structures for p 3 can be directly inferred from the 2-centered ones. Indeed, the 2-centered representation of spin s = J/2 of SL h (2, R) is in general replaced by the completely symmetric rank-J tensor representation 13 S J p of SL h (p, R). On the other hand, for p centers W (2.17) generally sits in the where ∧ 2 p is the rank-2 antisymmetric tensor representation 14 (which, in the case p = 2, becomes a singlet).
The same holds for SL(p, R). 14 In the case of GL(p, R), this is given by S λ (V ) (2.4) with V = R p =: p and λ := 1 2 ; see below (2.6). The same holds for SL(p, R). 15 This is generalized to V ⊗n (for a generic n) e.g. in [35], 9.3.1. 16 The weights/roots standard notation of irreps. is used throughout.
(2.26) Therefore, we obtained that and thus there are no ) when λ is a partition of an odd (positive) integer n.
In other words, there are no invariant polynomials in the fundamental representation V (λ 7 ) =: 56 of E 7 with an odd homogeneity degree, as also confirmed by the treatment of Sec. 2.3.1; more in general, this will hold at least for all the (simple and semi-simple) groups "of type E 7 " which we will consider : there are no invariant polynomials in the relevant irrep. R of G 4 with an odd homogeneity degree 17 .

Examples
We now consider explicit examples, relevant for p-centered (p 2) black holes in some D = 4 Maxwell-Einstein (super)gravity theories, with generalized electric-magnetic (U -)duality group G 4 ; as done above, we denote the relevant G 4 -representation in which the (fluxes of the) Abelian 2-form field strengths (and their duals) sit by 18 V = R, and we will specify it case by case.
In particular, we here consider the class of groups "of type E 7 " [15] which can be characterized as conformal groups of rank-3, simple Euclidean Jordan algebras J A 3 or J As 3 , or equivalently as the automorphism group of the Freudenthal triple system (FTS) M (J 3 ) constructed over such algebras [49]:

28)
A denotes the division algebras A = O, H, C, R, while A s denotes the corresponding split composition algebras A s = O s , H s , C s , R. The representation R pertains to M (J 3 ), and its dimension is 6q + 8, where the parameter q = dim R A (s) = 8, 4, 2, 1 for A (s) = O (s) , H (s) , C (s) , R, respectively. These class of groups "of type E 7 " has been recently studied as U -duality symmetries in the context of D = 4 locally supersymmetric theories of gravity in [17,18,19], as well as gauge (and global) symmetries in particular D = 3 gauge theories [42]. 17 The reason can be traced back to the fact that −I on R belongs to G4. For instance, it can be checked that the −I in the 56 of E7 preserves the symplectic metric C [M N] in 56 ⊗2 a and the quartic symmetric tensor . 18 It is worth pointing out that the irrep. R is real for the very non-compact real forms of G4 pertaining to the relevant U -duality groups, while usually for the other (non-compact) real forms it is pseudo-real (quaternionic). This reality property can e.g. be inferred from the corresponding (symmetric) embeddings into G3, the relevant U -duality symmetry in D = 3 space-time dimensions.
As an example, let us consider the fundamental representation R = 56 of E7 : it is real for the relevant non-compact real forms E 7(7) (split) and E 7(−25) (minimally non-compact), while it is pseudo-real (quaternionic) for E 7(−133) and E 7(−5) . Indeed, while E 7(7) and E 7(−25) respectively embed into E 8 (8) and E 8(−24) through a SL(2, R) commuting factor: E 7(−133) and E 7(−5) embed into E 8(−24) and E 8 (8) through an SU (2) factor: (2); 2 (m or n = 2) 4 (m or n = 6) 0 otherwise Table 1: Simple and semi-simple, non-degenerate U -duality groups G 4 "of type E 7 " [15]. The relevant symplectic irrep. R of G 4 is also reported. Note that the G 4 related to split composition algebras O s , H s , C s is the maximally non-compact (split) real form of the corresponding compact Lie group. The corresponding scalar manifolds are the symmetric spaces G 4 H 4 , where H 4 is the maximal compact subgroup (with symmetric embedding) of G 4 . The number of supercharges N of the resulting supergravity theory in D = 4 is also listed. The D = 5 uplift of the t 3 model (based on J 3 = R) is the pure N = 2, D = 5 supergravity. J H 3 is related to both 8 and 24 supersymmetries, because the corresponding supergravity theories share the very same bosonic sector [47,48,13].
An exception is provided by the stu model [33] (Sec. 2.3.5), whose triality symmetry is exploited within a particular case in Sec. 4.
From Sec. 2.2, it is here worth recalling that in general there are no polynomial invariants of (p, R) This is the prototypical case of groups "of type E 7 " [15]. In supergravity, this is related to the D = 4 theories with symmetric scalar manifold, based on the FTS M J O (2.29) 19 In LiE, one first increases the maximal size by typing the command "maxobjects 99999999".
The " [1]" at the end corresponds to the lowest representation. The output of the command is an integer, which we denote by d, times indicates the representation with highest weight b 1 λ 1 + · · · + b 7 λ 7 , the λ i being the fundamental weights (i = 1, ..., 7). If all b i 's are zero, then one has found polynomial invariants of homogeneity degree pa in p dim R = 56p variables; the real dimension of the vector space of such invariants is given by (recall (2.15)) By perusing the first few a's for the first few p's, one gets the following table 20 : (throughout the treatment, the blank entries are seemingly not accessible with the computing facilities available to us.) In the 2-centered case (p = 2), dim S 1 2 (56) E 7 = 1 corresponds to W (2.17). The interpretation of the other results is as follows: : where the 2-centered polynomial invariants 21 I 6 (degree 6), Tr I 2 (degree 8) and Tr I 3 (degree 12) have been firstly introduced in [32], and then studied in this very case in [22]. is a complete basis for the ring of polynomial invariants of (2, 56) of SL h (2, R) × E 7 , and it is finitely generating, namely all higher order polynomial invariants are simply polynomials in the polynomials of the set (2.33) itself [9]. 20 The result dim [S λ=0 p (V )] G 4 = 1 always trivially refers to a numerical constant. 21 As discussed at the end of Sec. 3 of [22], for p centers I6, as W (2.17), generally sits in the In the 3-centered case (p = 3), Table (2.31) yields that there are no E 7 -invariants for the partitions λ = 1 3 , 2 3 , 3 3 and hence there are no polynomial invariants of (3, 56) of SL h (3, R) × E 7 with homogeneity degree ≤ 10. The lowest possible degree is 12, at which Lie finds 5 invariants. The absence of an invariant corresponding to the partition λ = 1 3 , i.e. of a "3-centered analogue" of W (2.17) can be explained by the fact that 1 / ∈ 56 ⊗3 a (as mentioned, no invariant polynomials in the 56 of E 7 with an odd homogeneity degree exist at all). Then, one invariant of degree 18, and as many as 46 invariants of degree 24, are found.
In the 4-centered case (p = 4), there is an E 7 -invariant of degree 4 (the lowest possible degree). It can be regarded as the "4-centered analogue" of W (2.17), whose existence can be explained by the fact that ∃!1 ∈ 56 ⊗4 a , given by the complete antisymmetrization of the product of two symplectic metrics C M N of 56, such that (a = 1, ..., 4, M = 1, ..., 56) In the 5-centered case (p = 5), there are no invariants of degree ≤ 15, since the partitions λ = 1 5 , 2 5 and 3 5 do not yield any invariant for E 7 . Once again, the absence of an invariant corresponding to the partition λ = 1 5 , i.e. of a "5-centered analogue" of W (2.17), can be explained by the fact that 1 / ∈ 56 ⊗5 a . Finally, for the p = 6 and 8 -centered cases, we see that there is a unique polynomial invariant of (p, 56) of SL h (p, R) × E 7 (corresponding to the partition λ = 1 p ); again, for p = 6 and 8 it can be regarded as the "p-centered analogue" of W (2.17), whose existence can be explained by the fact that ∃!1 ∈ 56 ⊗6 a and ∃!1 ∈ 56 ⊗8 a , given by the complete antisymmetrization of the product of p = 6, 8 symplectic metrics C M N of 56, such that dim S 1 6 (56) E 7 = 1 :   ); thus, the structure of the ring of polynomial invariants of (2, 14 ′ ) of SL h (2, R) × Sp(6, R) is the very same as the one of (2, 56) of SL h (2, R) × E 7 . The same will hold for all other examples of groups "of type E 7 " relevant to D = 4 supergravity which we will consider below, meaning that the structure of two-centered invariants, as well as their interpretation (2.32), is the very same in all these cases.
However, this does not hold any more already starting from the 3-centered case (p = 3), as it is immediate to realize by comparing the p = 3 rows of (2.31) and (2.38). Indeed, Table (2.38), as Table  (2.31), yields that there are no Sp(6, R)-invariants for the partitions λ = 1 3 , 2 3 , 3 3 and hence there are no polynomial invariants of (3, 14 ′ ) of SL h (3, R) × Sp(6, R) with homogeneity degree ≤ 10, the lowest possible degree being 12, at which however Lie finds 4 invariants, instead of 5 invariants as in the E 7 case treated above. As above, the absence of an invariant corresponding to the partition λ = 1 3 , i.e. of a "3-centered analogue" of W (2.17), can be explained by the fact that 1 / ∈ 14 ′⊗3 a . In this case, the relevant SO(12)-representation is R = 32 or R = 32 ′ , namely one of the two chiral spinor representations. The dimension dim S (a p ) (32 (′) ) SO (12) for the partition λ = a p , yielding the (real) dimension of the vector space of polynomial invariants of homogeneity degree pa in p dim R = 32p variables, is given as above: dim S λ=a p 32 (′) SO (12) = dim S pa p, 32 (′) SL h (p,R)×SO(12) =: d.  (nonsupersymmetric theory, with G 4 = SL(6, R) [52]), where J C 3 and J Cs 3 are rank-3 Euclidean Jordan algebras over the complex numbers C and split complex numbers C s , respectively 23 .
In this case, the relevant SU (6)-representation is R = ∧ 3 6 =: 20, namely the rank-3 completely antisymmetric representation, built out from the fundamental representation 6. Due to the existence of the invariant ǫ-tensor in the 6 of SU (6), the irrep. 20 is real. The dimension dim S (a p ) (20) SU (6) for the partition λ = a p , yielding the (real) dimension of the vector space of polynomial invariants of homogeneity degree pa in p dim R = 20p variables, is given as above: Considerations essentially analogous to the previous cases hold in this case, as well. We now consider the so-called N = 2 stu model [33], whose U -duality group is G 4 = SL(2, R) × SO(2, 2) ∼ = SL(2, R) 3 , with the relevant BH flux representation being the tri-fundamental R = (2, 2, 2). This provides an example of group "of type E 7 " [15] different from the ones treated above. Indeed, SL(2, R) 3 can still be characterized as a conformal symmetry, but of a semi-simple, rank-3 Jordan algebra, namely J 3 = R ⊕ R ⊕ R, or equivalently as the automorphism group of the FTS M (J 3 ) constructed over such an algebra: (2.43) Actually, by virtue of the isomorphism R ⊕ R ⊕ R ∼ R ⊕ Γ 1,1 , this case can be regarded as the (m, n) = (2, 2) element of the infinite sequence of semi-simple rank-3 Jordan algebras R ⊕ Γ m−1,n−1 , where Γ m−1,n−1 denotes the Clifford algebra of O (m − 1, n − 1) [53]. This sequence can be related to D = 4 supergravity theories (displaying symmetric scalar manifolds) for m(or equivalently n)= 2 (N = 2) or 6 (N = 4). A complete basis of minimal degree (which turns out to be finitely generating [9]) of 2-centered BH invariant polynomials have been firstly determined in [32], and then further analyzed in [23] and [24]. The dimension dim S (a p ) ((2, 2, 2)) SL(2,R) 3 for the partition λ = a p , yielding the (real) dimension of the vector space of polynomial invariants of homogeneity degree pa in p dim R = 8p variables, is given as above: dim [S λ=a p ((2, 2, 2))] SL(2,R) 3 = dim [S pa (p, 2, 2, 2)] SL h (p,R)×SL(2,R) 3 =: d. We observe that the p = 2 row of  24 , corresponding to the invariance under the exchange of the three fundamentals 2's in R = (2, 2, 2), achieved by imposing an invariance under the symmetric group S 3 acting on the three 2's in R.
The implementation of the triality symmetry will be explicitly worked out in Sec. 4 for the case of p = 3 and a = 4, namely for the vector space of 3-centered invariant polynomials of degree 12, which, from Table ( Our analysis can be refined as follows : by looking directly for the (SL h (2, R) × G 4 )-invariants as above, we now consider the G 4 -invariants in S k ((R 2 ) ⊗ R). The formula (2.12) shows that these coincide with the G 4 -invariants in S λ (R), tensored by the SL h (2, R)-representation S λ (R 2 ), where λ ⊢ k and ht(λ) ≤ 2. By specifying this for the stu model, as done in all cases above, in Lie one types, for the partition k = a + b with a ≥ b, the following command (cfr. e.g. In the 2-centered case (p = 2), an S 3 -symmetric analysis of SL(2, R) 3 -and SL h (2, R) × SL(2, R) 3invariant homogeneous polynomials for 2-centered BHs in the stu model has been performed in [32,23,24],whereas an S 4 -symmetric treatment consistent in connection with the quantum entanglement of four qubits was given in [34].
Indeed, the relevant 2-centered representation for stu model is actually a quadri-fundamental : for p = 2 centers, one considers the invariants of the group SL h (2, R) × SL(2, R) 3 in the representation (2, 2, 2, 2). Thus, one may promote the S 3 -invariance (triality) to an invariance (tetrality) under the symmetric group S 4 acting on the four fundamentals 2's in (2, 2, 2, 2). A complete, minimal degree basis for the ring of SL h (2, R) × SL(2, R) 3 -invariant homogeneous polynomials is given by W, together with 2 quartic polynomials and with a sextic one, denoted by 25 I ′ 6 [34]. 24 The relevance of this symmetry to the theory of Quantum Information, and in particular to the classification of the quantum entanglement of three (and four) qubits has been recently studied, exploiting techniques and results from the supergravity side, also in the context of the so-called BH/qubit correspondence [37,54,55]. 25 Indeed, there is a slight difference in the definition of the (SL h (2, R) × G4)-invariant I6 for the models of D = 4 (super)gravity based on simple J3's [22] with respect to the definition of (SL h (2, R) × G4)-invariant I ′ 6 for the models of D = 4 (super)gravity based on the semi-simple sequence J3,m,n := R ⊕ Γm−1,n−1 [32,24]; this is discussed in Sec. 3 of [23].
When considering 2-centered BH physics, one must discriminate between the "horizontal" symmetry SL h (2, R) [32] and the U -duality symmetry G 4 = SL(2, R) 3 , on which a triality must be implemented. Therefore, by down-grading S 4 (pertaining to four qubits in QIT) to S 3 (pertaining to 2-centered stu BHs), the consistent S 3 -invariant p = 2 counting performed in [32,23,24] yields that an invariant polynomial of degree 8 is no more generated by the previous ones, and a finitely generating [9] complete basis for the ring of SL h (2, R) × SL(2, R) 3 -invariant homogeneous polynomials is given by four elements of degree 2, 4, 6 and 8 [32].

Geometric Interpretation
In this section we consider the invariants for SL h (p) × G 4 in (R p ) ⊗ R =: (p, R) in the case that 26 p ≤ r := dim R. (3.1) Note that r is even whenever the symplectic invariant 2-form C M N in R ⊗2 a is non-degenerate (as we assume throughout the paper).
We start and recall some classical results (mainly referring to [35]), and then we discuss the associated geometrical interpretation in terms of Grassmannians.
The main result is the observation that the G 4 -representation S (a p ) (R) which, as discussed in Sec. 2, produces all invariants in S ap ((R p ) ⊗ R), can be identified with the representation of G 4 on the homogeneous polynomials of degree a in the Plücker coordinates of the p-planes in R. Each of these Plücker coordinates is an SL h (p)-invariant homogeneous polynomial of degree p in the p dim R = pr coordinates on (R p )⊗R. Thus, the G 4 -invariant polynomials homogeneous of degree a in these Plücker coordinates provide exactly the (SL h (p) × G 4 )-invariant homogeneous polynomials of degree ap which are the object of our investigation.

Grassmannians
Any tensor t in (R p ) ⊗ R can be written as a sum t = min(r,p) a=1 x a ⊗ y a , with x a ∈ R p , y a ∈ R. Let f 1 , . . . , f p be the standard basis of R p . Writing each x a = p i=1 x ai f i , and using the bilinearity of ⊗, one finds that for certain uniquely determined elements r i ∈ R.
Since any (SL h (p) × G 4 )-invariant F is obviously an (SL h (p) × {I})-invariant, it is firstly convenient to study the invariants of SL h (p) × {I}. To this end, we only consider the action of SL h (p) on the first factor of (R p ) ⊗ R, so we are actually dealing with the direct sum of r copies of the 26 In the case p > r, one can easily show that there are no non-trivial invariants. This can be realized e.g. as follows. One can write a tensor t as t = p i=1 fi ⊗ ri (see Eq. (3.2)). In the case p > r, it is however more convenient to choose a basis e1, ..., er of R, so that the same tensor can be rewritten as t = r j=1 vj ⊗ ej, for (uniquely determined) vectors vj ∈ R p .
For a generic t (to be precise, for t outside the closed subset of codimension > 1 of R p ⊗ R defined by the vanishing of r × r minors of the matrix with rows v1, ..., vr), the vectors v1, ..., vr are linearly independent. Thus, there exists an element A ∈ SL h (p, R) such that Avi = fi, where {fi} is the standard basis of R p . Therefore, under the action of fundamental representation R p =: p of SL h (p). In the case r ≥ p (3.1), the ring of invariants in this case is well understood. Fixing a basis e 1 , . . . , e r of R, this ring is generated by the determinants of the (p × p)-minors of the r × p matrix T := T t whose columns are the vectors r 1 , . . . , r p ( [35], 11.1.2).
Note that all invariants F vanish on the tensors t = p i=1 f i ⊗ r i such that the rank of the matrix T t is less than p, i.e. when the r i do not span a p-dimensional subspace of R; such tensors t are called unstable (i.e., not semi-stable) tensors for this action. The (geometric) quotient ((R p ) ⊗ R)//SL h (p) is the image of the quotient map π given by generators of the ring of invariants F ([35], 11.1.2): Note that ∧ p R =: R ⊗p a = S λ (R) (with partition λ = 1 p ) has basis e I = e i 1 ∧. . .∧e ip , with i 1 < . . . < i p , and therefore π(t) = t I e I (with I collectively denoting the indices i 1 < . . . < i p ), where t I is the determinant of the minor of T t formed by the rows i 1 , . . . , i p .
The image of the quotient map π (3.3) consists of the decomposable tensors in ∧ p R. This map, when restricted to stable points, is the lift to linear spaces of the Plücker map Gr(p, R) → P(∧ p R), where Gr(p, R) denotes the Grassmannian of p-planes in R (see Sec. 3

.1.3).
Let now F be an (SL h (p) × G 4 )-invariant. Since it is trivially an (SL h (p) × {I})-invariant, from the above reasoning F is a polynomial in the determinants of (p × p)-minors of T t . Therefore, all such invariants can be determined with a two-step approach 27 : 1] first, one identifies the space of such polynomials as a representation of G 4 ; 2] then, one finds the G 4 -invariants in that space.
Step 1 is actually well-known when one considers the space of such polynomials as a representation for the larger group GL(R) =: GL(r) (namely, within (3.1)) : as a GL(R)-representation, the space of polynomials, homogeneous of degree a in the (p × p)-minors of the p × r matrices, is S a p (R) ( [35], 11.1.2).
In order to find the (SL h (p) × G 4 )-invariants in (R p ) ⊗ R, it then suffices to find the G 4 -invariants in the representations S a p (R) (step 2). This conclusion was already reached in Sec. 2.2; however, the above discussion clarifies how a G 4 -invariant in S a p (R) produces a polynomial on (R p ) ⊗ R.
We are now going to reformulate this reasoning in a geometrical way.

From Tensors to Planes
In order to study p-centered BHs, for the case (3.1), one can use the Grassmannian Gr(p, R) of p-planes in R as follows.
Using the notation of Sec. 3.1.1, any tensor t in (R p ) ⊗ R can be written as t = p i=1 f i ⊗ r i , for certain uniquely determined elements r i ∈ R. It is here convenient to consider the dense open subset (3.4) such that the p vectors r 1 , . . . , r p span a p-dimensional subspace of R (the upperscript "0" denotes the absence of unstable points). This yields a map G to Gr(p, R) as follows: It is worth noting that the action of SL h (p) on R p merely changes the basis of W t , so the map G is SL h (p)-invariant. It is obviously also GL h (p)-invariant, so it is actually identifying more tensors than strictly necessary for our purposes. The map G (3.5), besides being injective, is obviously also surjective: indeed, given a p-plane W ⊂ R, one can choose a basis r 1 , . . . , r p , and then W = W t , where t = p i=1 f i ⊗ r i . Thus, one gets the following bijection In particular, any G 4 -invariant function on the Grassmannian Gr(p, R) of p-planes in R will yield an (SL h (p) × G)-invariant function on ((R p ) ⊗ R) 0 , which will eventually extend 28 to the whole relevant irrep. (R p ) ⊗ R.

The Plücker Map
As Gr(p, R) is (a real subset of) a projective variety, which is moreover a p (r − p)-dimensional homogeneous space: one can proceed as follows. Recall that the Plücker map P is defined as the embedding In particular, the composition P • G of this map with G (3.5) maps t to r 1 ∧ . . . ∧ r p . Fixing a basis e 1 , . . . , e r of R, one thus gets the basis e I = e i 1 ∧ . . . ∧ e ip , with i 1 < . . . < i p , of ∧ p R (cfr. below (3.3)). The Plücker coordinates of W t are defined as the (p × p)-minors of the r × p matrix T := T t with columns r 1 , . . . , r p . The action of the group GL h (R) can be represented on the space of global sections Γ(Gr(p, R), L) on a line bundle L over Gr(p, R). Working over the complex numbers and denoting by P ic (X) the Picard group of the variety X, let us recall that P ic (Gr (p, R)) is generated by a (very ample) line bundle L, whose global sections are the Plücker coordinates themselves. In fact, Γ(Gr(p, R), L) ∼ = ∧ p R, (actually the dual representation thereof, since the coordinates are linear maps on ∧ p R). The action of GL h (R) on R then induces an action on the Grassmannian Gr(p, R) and thus on the spaces of global sections Γ(Gr(p, R), L). By recalling that ∧ p R = S λ (R) with partition λ = 1 p (cfr. below (2.6)), Bott's theorem (see e.g. Furthermore, any global section of L ⊗a is a linear combination of products of a sections of L (and therefore the map S a Γ(L) → Γ(L ⊗a ) is surjective); in terms of representations, this simply amounts to the statement that S aλ is a summand of S ap (R). Thus, any section of L ⊗a is a homogeneous polynomial in the Plücker coordinates of degree a. Given a G 4 -invariant F ∈ S aλ (R) ∼ = Γ(Gr(p, R), L ⊗a ), it corresponds to a degree a homogeneous polynomial in the Plücker coordinates, defined by the map (recall (3.5) and (3.6)): Thus, the composition yields a (SL h (p) × G 4 )-invariant which extends to the whole (R p ) ⊗ R. This provides a geometrical explanation of the treatment of Sec. 2, and in particular of the fact that the S λ (R) with λ = a p contribute to -and actually are the unique responsible for -the (SL h (p) × G 4 )-invariant homogeneous polynomials in (R p ) ⊗ R.
To summarize, in order to find (SL h (p) ⊗ G 4 )-invariant homogeneous polynomials F in the representation (R p ) ⊗ R, one needs to find invariant polynomialsF for the induced action of G 4 on ∧ p R: where In particular, if an invariant F is a homogeneous polynomial of degree k in the coefficients c ij of t = c ij f i ⊗ e j , then, as each Plücker coordinate is homogeneous of degree p in the c ij ,F is homogeneous of degree k/p in the Plücker coordinates. Thus, k must be a multiple of p. This matches the statement made below (2.15), and it is not surprising, as SL(p, C) contains the diagonal matrices ωI where ω = e 2πi/p and these act by multiplication by ω d on polynomials F of degree k; so, if F is SL h (p)-invariant, k must indeed be a multiple of p. Moreover, these invariantsF should be non-zero when restricted to the (semi-)stable decomposable tensors.
From the treatment of Secs. 2 and 3, as well as from Table (2.45), such 3-centered invariant polynomials lie in S 4 3 ((2, 2, 2)). In the present Section, we will determine a basis for their 10-dimensional space. Then, in Subsubsec. 4.3.4 we will implement invariance (triality) under the S 3 symmetric group acting on the three 2's in R, obtaining a basis of the resulting 4-dimensional vector space of S 3 × SL h (3, R) × SL (2, R) 3 -invariant homogeneous polynomials of degree 12 in the (3, 2, 2, 2), thus pertaining to the description of 3-centered BHs in the stu model.
From the treatment above, it clearly follows that the degree-12 homogeneous (SL h (3, R) × G 4 )invariant polynomial F 0 (4.11) can be consistently defined for all groups G 4 "of type E 7 ", and in particular at least for the class relevant to D = 4 supergravity theories with symmetric scalar manifolds, listed in Table 1. 4.2 Other Invariants from S 2 (S 2 3 ((2, 2, 2))) As a natural next step, one can try to determine other SL (2, R) 3 -invariants of degree 12 from quadratic invariants in S 2 3 ((2, 2, 2)).

The
The highest weight vector v ∈ V (a 1 , a 2 , a 3 ) satisfies Thus, V (a 1 , a 2 , a 3 ) can be realized as the vector space spanned by certain combinations of powers of lowering operators X i 's on its highest weight vector v itself: (4.26) By virtue of (4.25), the vector Y k is again an eigenvector of all three H i 's with weight (a 1 − 2k, a 2 − 2l, a 3 − 2m).
We are now going to exploit this general description in order to explicitly construct the 10 SL h (3, R) × SL (2, R) 3 -invariant homogeneous polynomials of degree 12 in the (3, 2, 2, 2) considered in Secs. 4.1 and 4.2, which constitute a complete basis for the corresponding 10-dimensional vector space resulting from Table (2.45).

stu Triality
In order to determine the remaining relevant 6 invariants of degree 12, one can now use the action of the symmetric group S 3 on R = (2, 2, 2) by permuting the tensor components, so (12) ∈ S 3 will map x abc to x bac , etc.. Consequently, S 3 will also act on the c ij 's, as well as on the Plücker coordinates p ijk . As we will see below, in the context of stu black holes, the invariance under S 3 must be enforced, because it corresponds to the triality symmetry [33] exhibited by such a model of N = 2, D = 4 supergravity.
Using this action, the 3 invariants just found in Sec. 4.3.3 give rise to the required set of 9 invariants. Including the invariant from Sec. 4.1 (which, as mentioned above, matches the one obtained in Sec. 4.3.2), one gets a total of 10 invariants of degree 12 in the c ij 's.