1 Introduction

Polynomial approximation from samples on manifolds and homogeneous spaces has already been extensively studied, cf. [18, 22, 28, 37] and references therein. Constituting distinct sampling rules, the concepts of cubatures and designs have also been widely investigated, cf. [20, 24, 25, 32, 36, 40, 41], where polynomials are integrated exactly by a finite sum over the sampling values. However, many open questions remain when dealing with polynomials on unions of nonconnected manifolds.

Orthogonal projectors with fixed rank are used in many applications for analysis and dimension reduction purposes, cf. [31, 44], leading to a function approximation problem on a single Grassmannian manifold. Projections with varying target dimensions are more flexible and may offer a larger range of applications. Therefore, we shall study unions of Grassmannians.

By studying the structure of polynomial spaces on the union of Grassmannians, some of our findings generalize results in [34]. In particular, we shall verify that the multiplicities of the irreducible representations of the orthogonal group occurring in an orthogonally invariant reproducing kernel Hilbert space on unions of Grassmannians coincide with the ranks of the kernel’s Fourier coefficients. This enables us to actually determine the multiplicities in the space of polynomials of degree t. Moreover, we construct the underlying reproducing kernels for the irreducible components. While cubatures and t-designs in single Grassmannians have been studied in [2,3,4,5], we shall also investigate these concepts in unions of Grassmannians. We derive a characterization as minimizers of an energy functional induced by a reproducing kernel. By numerically minimizing the energy functional, we compute candidates for t-designs, i.e., t-designs up to machine precision. We are then able to describe these candidates analytically and check that the energy functional is minimized exactly.

It should be mentioned that the topic shares some common themes with Euclidean designs, cf. [40], where unions of spheres with varying radii in Euclidean space are considered, see also [7,8,9,10]. The structure of the polynomial spaces on unions of spheres have been investigated in [21], but the ideas in those proofs do not work for unions of Grassmannians, whose structure appears to be more involved.

The outline is as follows. In Sect. 2 we recall some facts on polynomial spaces on single Grassmannians and their irreducible decompositions. Section 3 is dedicated to some elementary results on polynomial spaces on unions of Grassmannians. Direct consequences of irreducible decompositions of polynomials on symmetric matrices are studied in Sect. 4. In Sect. 5 we determine the multiplicities of the polynomial spaces on unions and construct the underlying reproducing kernels for the irreducible components. In Sect. 6 we introduce cubatures and t-designs on unions of Grassmannians and derive a characterization as minimizers of an energy functional induced by a reproducing kernel. We compute some analytical minimizers in Sect. 7.

2 Polynomials on Single Grassmannians

This section is dedicated to summarizing some facts about single Grassmannians, see, for instance, [5, 34]. The Grassmannian space of all k-dimensional linear subspaces of \({\mathbb {R}}^d\) is naturally identified with the set of orthogonal projectors on \({\mathbb {R}}^d\) of rank k denoted by

$$\begin{aligned} {\mathcal {G}}_{k,d} := \{ P \in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}} \;:\; P^{2}=P ;\; {{\mathrm{Tr}}}(P)=k \}, \end{aligned}$$

where \({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\) is the set of symmetric matrices in \({\mathbb {R}}^{d \times d}\). Each Grassmannian \({\mathcal {G}}_{k,d}\) admits a unique orthogonally invariant probability measure \(\sigma _{k,d}\) induced by the Haar (probability) measure \(\sigma _{{\mathcal {O}}(d)}\) on the orthogonal group \({\mathcal {O}}(d)\); i.e., for any \(Q \in {\mathcal {G}}_{k,d}\) and measurable function f,

$$\begin{aligned} \int _{{\mathcal {G}}_{k,d}} f(P) \mathrm {d}\sigma _{k,d}(P) = \int _{{\mathcal {O}}(d)} f(O Q O^{\top }) \mathrm {d}\sigma _{{\mathcal {O}}(d)}(O). \end{aligned}$$

The space of complex-valued, square-integrable functions \(L^{2}({\mathcal {G}}_{k,d})\), endowed with the inner product \((f, g)_{{\mathcal {G}}_{k,d}}\), decomposes into orthogonally invariant subspaces

$$\begin{aligned} L^{2}({\mathcal {G}}_{k,d}) = \bigoplus _{\ell (\lambda ) \le \min \{k,d-k\}}\!\!\!\!\!\!\! H_{\lambda }({\mathcal {G}}_{k,d}),\qquad H_{\lambda }({\mathcal {G}}_{k,d}) \perp H_{\lambda '}({\mathcal {G}}_{k,d}), \quad \lambda \ne \lambda ', \end{aligned}$$
(1)

where \(H_{\lambda }({\mathcal {G}}_{k,d})\) is equivalent to \(\mathcal H_{2\lambda }^{d}\), the irreducible representation of \({\mathcal {O}}(d)\) associated with the partition \(2\lambda =(2\lambda _{1},\ldots ,2\lambda _{t})\), cf. [5, 34]. Note that two representations are equivalent if there is a linear isomorphism that commutes with the group action. A partition of t is an integer vector \(\lambda =(\lambda _1,\ldots ,\lambda _t)\) with \(\lambda _1\ge \cdots \ge \lambda _t\ge 0\), \(|\lambda | = t\), where \(|\lambda |:=\sum _{i=1}^t\lambda _i\), and the length \(\ell (\lambda )\) is the number of nonzero parts of \(\lambda \). Note that we add and suppress zero entries in \(\lambda \) without further notice, so that we can also compare partitions of different lengths. For partitions \(\lambda ,\lambda '\) of integers \(t,t'\), respectively, we write \(\lambda \le \lambda '\) if and only if \(\lambda _i\le \lambda _i'\) for all \(i=1,\ldots ,\ell (\lambda )\).

The space of polynomials of degree at most t on \({\mathcal {G}}_{k,d}\) is given by

$$\begin{aligned} {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{k,d}):= \{ f|_{{\mathcal {G}}_{k,d}}: f\in {\mathbb {C}}[X]_t\}, \end{aligned}$$

where \({\mathbb {C}}[X]_t\) is the set of polynomials of degree at most t in \(d^{2}\) many variables arranged as a matrix \(X\in {\mathbb {C}}^{d\times d}\), and \(f|_{{\mathcal {G}}_{k,d}}\) denotes the restriction of f to \({\mathcal {G}}_{k,d}\). This polynomial space decomposes into

$$\begin{aligned} {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{k,d}) = \!\!\!\!\!\!\bigoplus _{{\begin{matrix} |\lambda | \le t,\\ \ell (\lambda ) \le \min \{k,d-k\} \end{matrix}}} \!\!\!\!\!\!H_{\lambda }({\mathcal {G}}_{k,d}), \end{aligned}$$

so that its dimension is calculated by adding the dimensions of each of the occurring \({\mathcal {H}}_{2\lambda }^{d}\). The dimension of \({\mathcal {H}}_{2\lambda }^{d}\) is specified in [27, Formulas (24.29) and (24.41)].

3 Polynomials on Unions of Grassmannians

Given a nonempty set \({\mathcal {I}}\subset \{1,\ldots ,d-1\}\), the corresponding union of Grassmannians is

$$\begin{aligned} {\mathcal {G}}_{{\mathcal {I}},d} :=\bigcup _{k\in {\mathcal {I}}} {\mathcal {G}}_{k,d} = \{ P \in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}} \;:\; P^{2}=P ,\; {{\mathrm{Tr}}}(P) \in {\mathcal {I}}\},\qquad {\mathcal {G}}_{d}:=\bigcup _{k=1}^{d-1} {\mathcal {G}}_{k,d}. \end{aligned}$$

An orthogonally invariant measure on \({\mathcal {G}}_{{\mathcal {I}},d}\) is derived by the sum of the corresponding measures on the single Grassmannians. In Sect. 6, we shall also allow for weighted sums. According to (1), the corresponding space of complex-valued, square-integrable functions \(L^{2}({\mathcal {G}}_{{\mathcal {I}},d})\) decomposes into

$$\begin{aligned} L^{2}({\mathcal {G}}_{{\mathcal {I}},d}) = \bigoplus _{\lambda \in \Lambda ^d_{\mathcal {I}}} H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d}), \qquad H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d}) \cong |{\mathcal {I}}^d_\lambda |{\mathcal {H}}_{2\lambda }^{d}, \end{aligned}$$
(2)

where \(\Lambda ^d_{\mathcal {I}}\) is the set of all partitions \(\lambda \) of length at most \(\max _{k \in {\mathcal {I}}}( \min (k, d-k))\) and the multiplicities are the cardinality of

$$\begin{aligned} {\mathcal {I}}^d_\lambda := \{ k \in {\mathcal {I}} : \ell (\lambda ) \le \min (k,d-k) \}. \end{aligned}$$

As for a single Grassmannian, we consider polynomials on \({\mathcal {G}}_{{\mathcal {I}},d}\) given by multivariate polynomials in the matrix entries of a given projector \(P \in {\mathcal {G}}_{{\mathcal {I}},d}\), i.e.,

$$\begin{aligned} {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d}) := \{ f|_{{\mathcal {G}}_{{\mathcal {I}},d}} : f\in {\mathbb {C}}[X]_t \}. \end{aligned}$$

This space decomposes orthogonally into

$$\begin{aligned} {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d}) =\bigoplus _{|\lambda |\le t,\;\lambda \in \Lambda ^d_{\mathcal {I}}} H_{\lambda }^{t}({\mathcal {G}}_{{\mathcal {I}},d}), \qquad H_{\lambda }^{t}({\mathcal {G}}_{{\mathcal {I}},d}) \cong \mu _{\lambda }^{d}({\mathcal {I}},t) {\mathcal {H}}_{2\lambda }^{d}, \end{aligned}$$

where the multiplicities \( \mu _{\lambda }^{d}({\mathcal {I}},t)\) still need to be determined. Indeed, this is the topic of the first part of the present paper.

Theorem 3.1

For \(t \ge 0\) and \({\mathcal {I}} = \{ k_i\}_{i=1}^r \subset \{1,\ldots ,d-1\}\) with \(r=|{\mathcal {I}}|\), the multiplicity of \({\mathcal {H}}^d_{2\lambda }\) in the direct sum \(\bigoplus _{i=1}^{s} {{\mathrm{Pol}}}_{t-i+1}({\mathcal {G}}_{k_{i},d})\) is a lower bound for that in \( {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\), where \(s:=\min \{t+1,|{\mathcal {I}}|\}\).

Proof

The cases \(t=0\) and \(|{\mathcal {I}}|=1\) are trivially fulfilled. Suppose \(t\ge 1\) and \(r\ge 2\). The restriction mapping \(|_{{\mathcal {G}}_{k_1,d}}: {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\rightarrow {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{k_1,d})\) is orthogonally invariant and surjective, so that \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) is equivalent to \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{k_1,d}) \oplus {{\mathrm{null}}}(|_{{\mathcal {G}}_{k_1,d}})\). Let \({{\mathrm{Tr}}}(\cdot )\) denote the trace. Since \(({{\mathrm{Tr}}}(\cdot ) - k_1) {{\mathrm{Pol}}}_{t-1}({\mathcal {G}}_{{\mathcal {I}},d}) \) is equivalent to \({{\mathrm{Pol}}}_{t-1}({\mathcal {G}}_{\{k_2,\ldots ,k_r\},d})\) and

$$\begin{aligned} ({{\mathrm{Tr}}}(\cdot ) - k_1) {{\mathrm{Pol}}}_{t-1}({\mathcal {G}}_{{\mathcal {I}},d}) \subset {{\mathrm{null}}}(|_{{\mathcal {G}}_{k_1,d}}), \end{aligned}$$

the number of irreducible components \({\mathcal {H}}^{d}_{2\lambda }\) in \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) is bigger than or equals the respective number in \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{k_1,d})\oplus {{\mathrm{Pol}}}_{t-1}({\mathcal {G}}_{\{k_2,\ldots ,k_r\},d})\). An induction over t and r completes the proof. \(\square \)

If we order \({\mathcal {I}} =\{ k_i\}_{i=1}^r\) by \( \min \{k_{1},d-k_{1}\} \ge \cdots \ge \min \{k_{r},d-k_{r}\}\), then counting the actual occurrences of \({\mathcal {H}}_{2\lambda }^d\) in \(\bigoplus _{i=1}^{s} {{\mathrm{Pol}}}_{t-i+1}({\mathcal {G}}_{k_{i},d})\) yields the following explicit lower bound.

Corollary 3.2

For \(t \ge 0\) and \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\), the multiplicities \( \mu _{\lambda }^{d}({\mathcal {I}},t)\) satisfy

$$\begin{aligned} \mu ^d_{\lambda }({\mathcal {I}},t) \ge \min \{ t - |\lambda |+1,|{\mathcal {I}}^d_\lambda |\}, \quad 0\le |\lambda |\le t, \quad \lambda \in \Lambda ^d_{\mathcal {I}}. \end{aligned}$$
(3)

Reformulation yields \(\mu ^d_{\lambda }({\mathcal {I}},|\lambda |+s) \ge \min \{ s+1,|{\mathcal {I}}^d_\lambda |\}\) for \(s\ge 0\). Due to (2), the upper bound \(\mu _{\lambda }^{d}({\mathcal {I}},t)\le |{\mathcal {I}}^d_\lambda |\) holds. Subsequent sections shall reveal that equality holds in (3). However, this requires a closer look at relations among irreducible representations and their reproducing kernels; see Appendix A.1 for some basics on reproducing kernels that shall be used in the following.

4 Determining the Multiplicities for a Few Special Cases

The space of polynomials on \({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\) of degree at most t and its subspace of homogeneous polynomials of degree t are denoted by

$$\begin{aligned} {{\mathrm{Pol}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})&: = \{ f|_{{\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}} : f\in {\mathbb {C}}[X]_t\},\\ {{\mathrm{Hom}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})&:=\{f\in {{\mathrm{Pol}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) : f(\alpha X)=\alpha ^t f(X), \; \alpha \in {\mathbb {R}},\;X\in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\}, \end{aligned}$$

respectively. The differential inner product between \(f,g \in {{\mathrm{Pol}}}_{t}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) given by

$$\begin{aligned} (f,{\overline{g}})_{\mathrm D} := f(D) {\overline{g}}(0),\qquad \text {where } \mathrm D:= \Big ( \frac{1}{2} ( 1 + \delta _{i,j} ) \partial _{i,j} \Big )_{i,j=1,\ldots ,d}\;, \end{aligned}$$

is orthogonally invariant, cf. [43], inducing the orthogonal decomposition

$$\begin{aligned} {{\mathrm{Pol}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) = \bigoplus _{s=0}^t {{\mathrm{Hom}}}_s({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}). \end{aligned}$$

Remark 4.1

In accordance with [30], the mapping \((X,Y)\mapsto \frac{1}{s!}{{\mathrm{Tr}}}(XY)^s\) is the reproducing kernel for \({{\mathrm{Hom}}}_s({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) with respect to the differentiation inner product, cf. also Appendix A.1.

Let \({{\mathrm{GL}}}(d)\) denote the general linear group. The space \({{\mathrm{Hom}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \) decomposes orthogonally into subspaces \(F_\lambda ({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) invariant under the action \(f \mapsto f(L \cdot L^{\top })\), for \(L\in {{\mathrm{GL}}}(d)\), by

$$\begin{aligned} {{\mathrm{Hom}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) = \bigoplus _{|\lambda |=t,\;\;\ell (\lambda )\le d} F_\lambda ({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}),\qquad F_\lambda ({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\cong {\mathcal {F}}_{2\lambda }^{d}, \end{aligned}$$
(4)

where \({\mathcal {F}}_{2\lambda }^{d}\) is the irreducible representation of \({{\mathrm{GL}}}(d)\) associated with \(2\lambda =(2\lambda _{1},\ldots ,2\lambda _{t})\), cf. [30].

Remark 4.2

One can check that the function \((X,Y)\rightarrow \frac{1}{|\lambda |!} C_\lambda (X Y)\) is the reproducing kernel for \(F_\lambda ({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) with respect to the differentiation inner product, where \(C_\lambda \) is the zonal polynomial of index \(\lambda \), cf. [30] and Appendix B.1.

By restricting the group action from \({{\mathrm{GL}}}(d)\) to \({\mathcal {O}}(d)\), the space \(F_\lambda ({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) decomposes further as

$$\begin{aligned} F_{\lambda }({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \cong {\mathcal {F}}_{2\lambda }^{d} \cong \bigoplus _{\lambda '\le 2\lambda }\nu ^d_{2\lambda ,\lambda '} {\mathcal {H}}^d_{\lambda '},\qquad \ell (\lambda ) \le d, \end{aligned}$$
(5)

where the multiplicities \(\nu ^d_{2\lambda ,\lambda '}\in {\mathbb {N}}_0\) are determined by the corresponding branching rule, cf. [26, 33, 35]. If \(\lambda '\) is such that \({\mathcal {H}}^d_{\lambda '}\) is not defined, then we simply put \(\nu ^d_{2\lambda ,\lambda '}=0\). We note that

$$\begin{aligned} \nu ^d_{2\lambda ,(0)}=1 \quad \text {for }\ell (\lambda )\le d,\qquad \qquad \nu ^d_{2\lambda ,2\lambda }=1 \quad \text {for }\ell (\lambda ) \le d/2. \end{aligned}$$

Thus, we obtain \(\mu _{\lambda }^{d}({\mathcal {I}},t) = 1\), for \(|\lambda |=t\ge 0\) with \(\lambda \in \Lambda ^d_{\mathcal {I}}\). The latter enables us to verify that the lower bounds in Corollary 3.2 are an equality for \(|{\mathcal {I}}|=2\). Indeed, if we order \({\mathcal {I}} = \{ k_{1}, k_{2} \} \subset \{1,\ldots ,d-1\}\) by \(\min \{k_{1},d-k_{1}\} \ge \min \{k_{2},d-k_{2}\}\), then Theorem 3.1 and \(\mu _{\lambda }^{d}({\mathcal {I}},t) = 1\) imply

$$\begin{aligned} \mu _{\lambda }^{d}(\{k_1,k_2\},t) = {\left\{ \begin{array}{ll} 1 , &{} |\lambda |=t \text { with } \ell (\lambda )\le \min \{k_{1},d-k_{1}\},\\ 1 , &{} |\lambda |\le t-1 \text { with } \min \{k_{2},d-k_{2}\}<\ell (\lambda )\le \min \{k_{1},d-k_{1}\},\\ 2,&{} |\lambda |\le t-1 \text { with } \ell (\lambda )\le \min \{k_{2},d-k_{2}\}, \end{array}\right. } \end{aligned}$$

which means that equality holds in (3).

For arbitrary \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\), we can still use (5) to determine the multiplicities \(\mu _\lambda ^d({\mathcal {I}},t)\) provided that t is sufficiently small, but this takes some preparation. The space of homogeneous polynomials restricted to \({\mathcal {G}}_{{\mathcal {I}},d}\) is denoted by \( {{\mathrm{Hom}}}_t({\mathcal {G}}_{{\mathcal {I}},d}) := \{ f|_{{\mathcal {G}}_{{\mathcal {I}},d}} : f\in {{\mathrm{Hom}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \}. \) It is important to notice that the restriction of \({{\mathrm{Hom}}}_t({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) to \({\mathcal {G}}_{{\mathcal {I}},d}\) yields (almost) the entire space \({{\mathrm{Pol}}}_t({\mathcal {G}}_{{\mathcal {I}},d})\):

Theorem 4.3

For \(t\ge 0\) and \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\), the polynomial space decomposes into

$$\begin{aligned} {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})&= {\left\{ \begin{array}{ll} {{\mathrm{Hom}}}_t({\mathcal {G}}_{{\mathcal {I}},d})\oplus {{\mathrm{Hom}}}_0({\mathcal {G}}_{{\mathcal {I}},d}), &{}1\le t \le |{\mathcal {I}}|-1,\\ {{\mathrm{Hom}}}_t({\mathcal {G}}_{{\mathcal {I}},d}), &{}\text {else}. \end{array}\right. } \end{aligned}$$

Proof

First we note that \({{\mathrm{Tr}}}(X \cdot )^{t}\), \(X \in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\), linearly generates the space \({{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\), cf. Remark 4.1, (27), and (28). Now, for \(t\ge 1\) and \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\), we have

$$\begin{aligned} {{\mathrm{Tr}}}(XP)^{s}={{\mathrm{Tr}}}(XP)^{s-1}{{\mathrm{Tr}}}(XP^{t-s+1}), \qquad X \in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}},\quad P \in {\mathcal {G}}_{{\mathcal {I}},d}, \quad 1 \le s \le t. \end{aligned}$$

Since the term on the right-hand side is a homogeneous polynomial of degree t in P restricted to \({\mathcal {G}}_{{\mathcal {I}},d}\), we deduce

$$\begin{aligned} {{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d}) = \{ f|_{{\mathcal {G}}_{{\mathcal {I}},d}} : f\in {{\mathrm{Hom}}}_s({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}),\; s=1,\ldots ,t \}. \end{aligned}$$
(6)

Thus, it remains to check that \(1|_{{\mathcal {G}}_{{\mathcal {I}},d}} \not \in {{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) if and only if \(1\le t \le |\mathcal I|- 1\). Since \(1|_{{\mathcal {G}}_{{\mathcal {I}},d}}\) is orthogonally invariant, it is sufficient to consider the orthogonally invariant subspace of \({{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) denoted by \({{\mathrm{Hom}}}_{t}^{\mathcal O(d)}({\mathcal {G}}_{{\mathcal {I}},d})\). For \(f \in {{\mathrm{Hom}}}_{t}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\), we define

$$\begin{aligned} f_{\circ }(X) := \int _{{\mathcal {O}}(d)} f(O X O^{\top })d\mu _{\mathcal O(d)}(O),\qquad X \in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}, \end{aligned}$$

so that \({{\mathrm{Hom}}}_{t}^{{\mathcal {O}}(d)}({\mathcal {G}}_{{\mathcal {I}},d}) = {{\mathrm{span}}}\big \{ f_{\circ }|_{{\mathcal {G}}_{{\mathcal {I}},d}} : f \in {{\mathrm{Hom}}}_{t}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \big \}\). According to invariant theory, the ring of orthogonally invariant polynomials on \({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\) is generated by polynomials of the form \({{\mathrm{Tr}}}(X^{l})\), \(X \in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\), \({l} \in {\mathbb {N}}_{0}\), cf. [42, Theorem 7.1]. Since \(f_{\circ }\) is also homogeneous of degree t,

$$\begin{aligned} f_{\circ }(X)= \sum _{ {l}_{1}+\cdots +{l}_{t}=t } f_{{l}_{1},\ldots ,{l}_{t}}\prod _{i=1}^{t} {{\mathrm{Tr}}}(X^{{l}_{i}}),\quad X\in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}. \end{aligned}$$

For \(t\ge 1\), the restriction \(f_{\circ }|_{{\mathcal {G}}_{{\mathcal {I}},d}}\) is a linear combination of the functions \({{\mathrm{Tr}}}(\cdot )^{s}|_{{\mathcal {G}}_{{\mathcal {I}},d}}\), \(s=1,\ldots ,t\). According to (6), these functions are contained in \({{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\), so that \({{\mathrm{Hom}}}_{t}^{\mathcal O(d)}({\mathcal {G}}_{{\mathcal {I}},d})\) is spanned by \({{\mathrm{Tr}}}(\cdot )|_{{\mathcal {G}}_{{\mathcal {I}},d}}\), ..., \({{\mathrm{Tr}}}(\cdot )^t|_{{\mathcal {G}}_{{\mathcal {I}},d}}\), for \(t\ge 1\). The invertibility of the Vandermonde matrix implies that \(1_{{\mathcal {G}}_{{\mathcal {I}},d}}\) and \({{\mathrm{Tr}}}(\cdot )^{i}|_{{\mathcal {G}}_{{\mathcal {I}},d}}\), \(i=1,\ldots ,t\), are linearly independent if and only if \(1\le t \le |{\mathcal {I}}|- 1\). \(\square \)

Remark 4.4

Fix some \(t\ge 0\) and \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\). The map \( (P,Q)\mapsto {{\mathrm{Tr}}}(PQ)^t \) is a reproducing kernel for \({{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\). For any constant \(C > 0\), the map \( (P,Q)\mapsto {{\mathrm{Tr}}}(PQ)^t+C \) is a reproducing kernel for \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\).

Next, we determine the multiplicities \(\mu _\lambda ^d({\mathcal {I}},t)\), for \(t=1,2,3\), by deriving upper bounds that match the lower bounds in Corollary 3.2. The decompositions (4) and (5) yield \({{\mathrm{Hom}}}_{1}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \cong {\mathcal {H}}_{(0)}^{d} \oplus {\mathcal {H}}_{(2)}^{d}\), for \(d\ge 2\), and

$$\begin{aligned} {{\mathrm{Hom}}}_{2}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \cong {\left\{ \begin{array}{ll} 2{\mathcal {H}}_{(0)}^{d} \oplus 2{\mathcal {H}}_{(2)}^{d} \oplus \mathcal H_{(4)}^{d} \oplus {\mathcal {H}}_{(2,2)}^{d}, &{} d\ge 4, \\ 2{\mathcal {H}}_{(0)}^{d} \oplus 2{\mathcal {H}}_{(2)}^{d} \oplus \mathcal H_{(4)}^{d}, &{} d = 3,\\ 2{\mathcal {H}}_{(0)}^{d} \oplus \;\;{\mathcal {H}}_{(2)}^{d} \oplus {\mathcal {H}}_{(4)}^{d}, &{} d = 2.\\ \end{array}\right. } \end{aligned}$$
(7)

The multiplicities of \({\mathcal {H}}_\lambda ^d\) in \({{\mathrm{Hom}}}_{t}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\) are upper bounds for \(\mu ^d_{\lambda }({\mathcal {I}},t)\) since the restriction mapping is orthogonally invariant. For \(t=1,2\), the lower bounds in Corollary 3.2 are matched, so that we have determined \(\mu ^d_{\lambda }({\mathcal {I}},1)\) and \(\mu ^d_{\lambda }({\mathcal {I}},2)\) for any index set \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\). For \(t=3\), the analysis is more difficult, and we observe that the branching rules yield

$$\begin{aligned} {{\mathrm{Hom}}}_{3}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \cong {\left\{ \begin{array}{ll} 3{\mathcal {H}}_{(0)}^{d} \oplus 4{\mathcal {H}}_{(2)}^{d} \oplus 2{\mathcal {H}}_{(4)}^{d} \oplus 2{\mathcal {H}}_{(2,2)}^{d} \oplus {\mathcal {H}}_{(3,1)}^{d} \oplus {\mathcal {H}}_{(6)}^{d} \oplus \mathcal H_{(4,2)}^{d} \oplus {\mathcal {H}}_{(2,2,2)}^{d}, &{} d \ge 6,\\ 3{\mathcal {H}}_{(0)}^{d} \oplus 4 {\mathcal {H}}_{(2)}^{d} \oplus 2\mathcal H_{(4)}^{d} \oplus 2{\mathcal {H}}_{(2,2)}^{d} \oplus \mathcal H_{(3,1)}^{d} \oplus {\mathcal {H}}_{(6)}^{d} \oplus \mathcal H_{(4,2)}^{d}, &{} d = 5,\\ 3{\mathcal {H}}_{(0)}^{d} \oplus 4 {\mathcal {H}}_{(2)}^{d} \oplus 2\mathcal H_{(4)}^{d} \oplus \;\;{\mathcal {H}}_{(2,2)}^{d} \oplus \mathcal H_{(3,1)}^{d} \oplus {\mathcal {H}}_{(6)}^{d} \oplus \mathcal H_{(4,2)}^{d}, &{} d = 4,\\ 3{\mathcal {H}}_{(0)}^{d} \oplus 3 {\mathcal {H}}_{(2)}^{d} \oplus 2\mathcal H_{(4)}^{d} \qquad \qquad \,\,\oplus {\mathcal {H}}_{(3,1)}^{d} \oplus {\mathcal {H}}_{(6)}^{d}, &{} d = 3,\\ 2{\mathcal {H}}_{(0)}^{d} \oplus 2{\mathcal {H}}_{(2)}^{d} \oplus \;\;{\mathcal {H}}_{(4)}^{d} \qquad \qquad \qquad \qquad \,\oplus \mathcal H_{(6)}^{d}, &{} d = 2. \end{array}\right. } \end{aligned}$$

The multiplicity of \({\mathcal {H}}_{(2)}^{d}\) in \({{\mathrm{Hom}}}_{3}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \) does not match the lower bound in Corollary 3.2. Instead, we found that the kernel

$$\begin{aligned} \begin{aligned} K(X,Y) =&\frac{1}{d+2} \Big ({{\mathrm{Tr}}}(X^{2}Y^{2}) {{\mathrm{Tr}}}(X Y) - {{\mathrm{Tr}}}(X^{2}Y){{\mathrm{Tr}}}(XY^{2})\Big )\\&- \frac{1}{(3d+4)(d+2)} \Big ({{\mathrm{Tr}}}(X^{2}Y^{2}){{\mathrm{Tr}}}(X){{\mathrm{Tr}}}(Y) - {{\mathrm{Tr}}}(X^{2}Y){{\mathrm{Tr}}}(X){{\mathrm{Tr}}}(Y^{2}) \\&\quad \qquad \qquad \qquad \qquad \qquad - {{\mathrm{Tr}}}(X Y^{2}){{\mathrm{Tr}}}(X^{2}){{\mathrm{Tr}}}(Y) + {{\mathrm{Tr}}}(XY){{\mathrm{Tr}}}(X^{2}){{\mathrm{Tr}}}(Y^{2}) \Big ), \end{aligned} \end{aligned}$$
(8)

which may not have been observed in the literature yet, reproduces a subspace of \({{\mathrm{Hom}}}_{3}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \) equivalent to \({\mathcal {H}}^d_{(3,1)}\oplus {\mathcal {H}}^d_{(2)}\), for \(d>2\), and equivalent to \({\mathcal {H}}_{(2)}^{d}\), for \(d=2\), respectively. Since K vanishes on any Grassmannian, i.e., \(K(X,Y)=0\), for all \(Y\in {\mathcal {G}}_{k,d}\), \(k \in \{1,\ldots ,d-1\}\) and \(X\in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\), we deduce that the multiplicity of \({\mathcal {H}}_{(2)}^{d}\) in \({{\mathrm{Pol}}}_{3}({\mathcal {G}}_{{\mathcal {I}},d})\) is less than in \( {{\mathrm{Hom}}}_{3}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}) \). Now, for even partitions, the resulting upper bounds on the multiplicities match the lower bounds in Corollary 3.2 for \(t=3\) and any index set \({\mathcal {I}} \subset \{1,\ldots ,d-1\}\).

To determine the multiplicities of the irreducible subspaces of \({{\mathrm{Pol}}}_{3}({\mathcal {G}}_{{\mathcal {I}},d})\) for \(|{\mathcal {I}}|\ge 3\), we used the reproducing kernel K in (8). We shall further explore the reproducing kernels of the irreducible components to determine the multiplicities in \({{\mathrm{Pol}}}_t({\mathcal {G}}_{{\mathcal {I}},d})\) for \(|{\mathcal {I}}|\ge 3\) and \(t\ge 4\).

5 Zonal Kernels and Harmonic Analysis on Unions of Grassmannians

For \(\ell (\lambda )\le \min (k,d-k,l,d-l)\), the spaces \(H_{\lambda }({\mathcal {G}}_{k,d})\) and \(H_{\lambda }({\mathcal {G}}_{l,d})\) are equivalent; hence, there is a real intertwining isomorphism \(T_{\lambda }^{k,l}: H_{\lambda }({\mathcal {G}}_{k,d}) \rightarrow H_{\lambda }({\mathcal {G}}_{l,d})\). In particular, \(T_{\lambda }^{k,l}\) commutes with complex conjugation and the group action. It can be realized by an integral transform with a unique real-valued zonal function \(p_{\lambda }^{k,{l}}:{\mathcal {G}}_{k,d}\times {\mathcal {G}}_{{l},d}\rightarrow {\mathbb {R}}\), so that

$$\begin{aligned} T_{\lambda }^{k,{l}}f = \int _{{\mathcal {G}}_{{k},d}} p_{\lambda }^{k,{l}}(P,\cdot )f(P)\mathrm {d}\sigma _{k,d}(P). \end{aligned}$$
(9)

Note that zonal means \(p_{\lambda }^{k,{l}}(OPO^\top ,OQO^\top )=p_{\lambda }^{k,{l}}(P,Q)\), for \(O\in {\mathcal {O}}(d)\) and \(P\in {\mathcal {G}}_{{k},d}\), \(Q\in {\mathcal {G}}_{{l},d}\). For fixed \(1 \le k \le {l} \le \tfrac{d}{2}\) and \(\ell (\lambda )\le k\), the intertwining functions \( p_{\lambda }^{k,{l}}\) were studied in [34] and expanded into the zonal polynomials by

$$\begin{aligned} p_{\lambda }^{k,{l}}(P,Q)= & {} b_{\lambda }^{k,{l},d} \sum _{\lambda '\le \lambda } c_{\lambda ,\lambda '}^{\frac{d}{2}} q_{\lambda ,\lambda '}(\tfrac{k}{2}) q_{\lambda ,\lambda '}(\tfrac{l}{2}) C_{\lambda '}(PQ), \nonumber \\&\text { for }\ell (\lambda )\le k, \;\;1 \le k \le {l} \le \tfrac{d}{2}, \end{aligned}$$
(10)

where \(b_{\lambda }^{k,{l},d}\in {\mathbb {R}}\) is a scaling constant, \(c_{\lambda ,\lambda }^{\frac{d}{2}}=1\), and \(q_{\lambda ,\lambda '}\) is a polynomial of degree \(|\lambda |-|\lambda '|\) given by

$$\begin{aligned} q_{\lambda ,\lambda '}(x):= \prod _{i=1}^{m} (x-\tfrac{1}{2}(i-1)+\lambda '_{i})_{\lambda _{i}-\lambda '_{i}} = \frac{(x)_{\lambda }}{(x)_{\lambda '}}, \quad x \in {\mathbb {R}}. \end{aligned}$$

Potential zeros in the denominator of \(\frac{(x)_{\lambda }}{(x)_{\lambda '}}\) cancel out, so that the fraction is well defined. Up to the scaling, which we have not specified yet, the functions \(p_{\lambda }^{k,k}\) are the reproducing kernels for \(H_{\lambda }({\mathcal {G}}_{k,d})\) with respect to the \(L^2\) inner product when \(1\le k\le d/2\) and \(\ell (\lambda )\le k\), cf. [34].

The sum of the right-hand side in (10) is still well defined for all \(k,l\in \{1,\ldots ,d-1\}\) and \(\ell (\lambda )\le \min (k,d-k,l,d-l)\). One of our contributions going beyond [34] is to determine the reproducing kernel of \(H_{\lambda }^{|\lambda |}({\mathcal {G}}_{d})\) with the help of a particular extension of (10) to this broader range of parameters. Recall that \({\mathcal {G}}_{d}=\bigcup _{k=1}^{d-1} {\mathcal {G}}_{k,d}\).

Theorem 5.1

Let \(\ell (\lambda )\le d/2\). The reproducing kernel of \(H_{\lambda }^{|\lambda |}({\mathcal {G}}_{d})\) with respect to the \(L^2\) inner product is a multiple of

$$\begin{aligned} p_{\lambda }(P,Q) := \sum \limits _{\lambda ' \le \lambda } c_{\lambda ,\lambda '}^{\frac{d}{2}} q_{\lambda ,\lambda '}(\tfrac{1}{2} {{\mathrm{Tr}}}(P)) q_{\lambda ,\lambda '}(\tfrac{1}{2} {{\mathrm{Tr}}}(Q)) C_{\lambda '}(P Q),\quad P, Q \in {\mathcal {G}}_{d}. \end{aligned}$$

In order to verify Theorem 5.1, we choose a suitable normalization of the intertwining functions, induced by the following selection of intertwining operators.

Proposition 5.2

For all \(k,l\in \{1,\ldots ,d-1\}\) and \(\ell (\lambda )\le \min (k,d-k,l,d-l)\), there are real isometric isomorph intertwining operators \(T_{\lambda }^{k,{l}}:H_{\lambda }({\mathcal {G}}_{{k},d}) \rightarrow H_{\lambda }({\mathcal {G}}_{l,d})\) such that the following diagram commutes:

figure a

Proof

Let us fix an index s with \(\ell (\lambda )\le s\le d - \ell (\lambda )\). There are real isometric isomorph operators \(\tilde{T}_{\lambda }^{k,s}\), \(k \in \{\ell (\lambda ),\ldots , \ell (\lambda )\}\), which intertwine the spaces \(H_{\lambda }({\mathcal {G}}_{k,d})\) and \(H_{\lambda }({\mathcal {G}}_{s,d})\). We now define

$$\begin{aligned} T_{\lambda }^{k,{l}} := ({\tilde{T}}_{\lambda }^{{l},s})^*{\tilde{T}}_{\lambda }^{k,s},\qquad \ell (\lambda )\le k,{l}\le d-\ell (\lambda ), \end{aligned}$$

and straightforward calculations yield the statement. \(\square \)

The integral operators in Proposition 5.2 induce intertwining functions \(p_\lambda ^{k,{l}}\) via (9) satisfying

$$\begin{aligned} \dim ({\mathcal {H}}_{2\lambda }^{d})&=\int _{{\mathcal {G}}_{k,d}}\int _{{\mathcal {G}}_{{l},d}} |p_{\lambda }^{k,{l}}(P,Q)|^{2} \mathrm {d}\sigma _{{l},d}(Q) \mathrm {d}\sigma _{k,d}(P), \end{aligned}$$
(11)
$$\begin{aligned} p_{\lambda }^{k,{l}}(P,Q)&= \big (p_{\lambda }^{k,m}(P,\cdot ), p_{\lambda }^{m,{l}}(\cdot ,Q)\big )_{{\mathcal {G}}_{m,d}} \quad \text {for }m \in \{\ell (\lambda ),\ldots ,d-\ell (\lambda )\}. \end{aligned}$$
(12)

Remark 5.3

The intertwining functions \(p_{\lambda }^{k,k}\) are the reproducing kernels of \(H_\lambda ({\mathcal {G}}_{k,d})\cong {\mathcal {H}}^d_{2\lambda }\) with respect to the standard \(L^2\) inner product for \(k\in \{1,\ldots ,d-1\}\) and \(\ell (\lambda )\le \min (k,d-k)\).

Let \(P_{\lambda }^{k,{l}}:{\mathcal {G}}_{{\mathcal {I}},d}\times {\mathcal {G}}_{{\mathcal {I}},d} \rightarrow {\mathbb {R}}\) denote the zero extension of \(p_\lambda ^{k,{l}}\). It follows from [34] that the collection of zonal functions \(\{P_{\lambda }^{k,{l}} : \lambda \in \Lambda ^d_{\mathcal {I}};\; k,{l}\in {\mathcal {I}}^d_\lambda \}\) is an orthogonal basis for \(L^{2}_{{\mathcal {O}}(d)}({\mathcal {G}}_{{\mathcal {I}},d}\times {\mathcal {G}}_{{\mathcal {I}},d})\), the space of square integrable functions that are zonal. Hence, any zonal function \(f \in L^{2}_{\mathcal O(d)}({\mathcal {G}}_{{\mathcal {I}},d} \times {\mathcal {G}}_{{\mathcal {I}},d})\) can be expanded into a Fourier series; i.e.,

$$\begin{aligned} f = \sum _{\lambda \in \Lambda ^d_{\mathcal {I}} } \sum _{ k,{l} \in {\mathcal {I}}^d_\lambda } {\hat{f}}_{\lambda }^{k,{l}} P_{\lambda }^{k,{l}} = \sum _{\lambda \in \Lambda ^d_{\mathcal {I}}} {{\mathrm{Tr}}}( \hat{f}_{\lambda }^{\top } P_{\lambda }), \end{aligned}$$
(13)

where the Fourier coefficients \({\hat{f}}_{\lambda }^{k,{l}}\) and the basis functions \(P_{\lambda }^{k,{l}}\) are arranged in matrix form

$$\begin{aligned} {\hat{f}}_{\lambda } := \big ({\hat{f}}_{\lambda }^{k,{l}} \big )_{k,{l} \in {\mathcal {I}}^d_\lambda } \in {\mathbb {C}}^{|{\mathcal {I}}^d_\lambda |\times |{\mathcal {I}}^d_\lambda |}, \qquad P_{\lambda } := \big ( P_{\lambda }^{k,{l}} \big )_{k,{l} \in {\mathcal {I}}^d_\lambda }. \end{aligned}$$

Convolving two continuous zonal functions \(f,g \in L^{2}_{\mathcal O(d)}({\mathcal {G}}_{{\mathcal {I}},d} \times {\mathcal {G}}_{{\mathcal {I}},d})\),

$$\begin{aligned} (f * g) (P,Q) := \big (f (P,\cdot ), \overline{g (\cdot ,Q)}\big )_{{\mathcal {G}}_{{\mathcal {I}},d}}= \sum _{k \in \mathcal I}\int _{{\mathcal {G}}_{k,d}} \!\!\!\!\!\!f (P,R)g (R,Q)d\sigma _{k,d}(R), \end{aligned}$$

yields again a continuous zonal function \(f*g\). It is straightforward to check that its Fourier coefficients are \(\widehat{(f*g)}_\lambda = {\hat{f}}_{\lambda } {\hat{g}}_{\lambda } \in {\mathbb {C}}^{|{\mathcal {I}}^d_\lambda |\times |{\mathcal {I}}^d_\lambda |}\), \(\lambda \in \Lambda ^d_{\mathcal {I}}\).

Remark 5.4

This convolution property implies that the kernel \((P,Q) \mapsto {{\mathrm{Tr}}}(P_{\lambda }(P,Q))\), for \(\lambda \in \Lambda ^d_{\mathcal {I}}\), is the reproducing kernel of \(H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d})\) with respect to the \(L^{2}({\mathcal {G}}_{{\mathcal {I}},d})\) inner product.

The Fourier coefficients \({\hat{K}}_{\lambda } \in {\mathbb {C}}^{|{\mathcal {I}}^d_\lambda |\times |{\mathcal {I}}^d_\lambda |}\) of a positive definite zonal kernel \(K \in L^{2}_{\mathcal O(d)}({\mathcal {G}}_{{\mathcal {I}},d} \times {\mathcal {G}}_{{\mathcal {I}},d})\) are positive semidefinite matrices and thus allow for a spectral decomposition

$$\begin{aligned} {\hat{K}}_{\lambda } = \sum _{i=1}^{|{\mathcal {I}}^d_\lambda |} \alpha ^i_{\lambda } \hat{K}_{\lambda }^i,\qquad \lambda \in \Lambda ^d_{\mathcal {I}}, \end{aligned}$$
(14)

where \(\alpha ^1_{\lambda } \ge \cdots \ge \alpha ^{|{\mathcal {I}}^d_\lambda |}_{\lambda } \ge 0\) and \(\hat{K}_{\lambda }^i\in {\mathbb {C}}^{|{\mathcal {I}}^d_\lambda |\times |{\mathcal {I}}^d_\lambda |}\) are orthogonal rank-1 projectors corresponding to an eigenbasis of \(\hat{K}_\lambda \). The corresponding kernels

$$\begin{aligned} K^i_\lambda := {{\mathrm{Tr}}}(\hat{K}^{i\top }_\lambda P_{\lambda }),\quad i=1,\ldots ,|{\mathcal {I}}^d_\lambda |, \end{aligned}$$
(15)

are also positive definite. This spectral decomposition of K yields the irreducible decomposition of the underlying reproducing kernel Hilbert space denoted by \({\mathcal {S}}(K)\), see also (27):

Theorem 5.5

Let \(K : {\mathcal {G}}_{{\mathcal {I}},d} \times {\mathcal {G}}_{{\mathcal {I}},d} \rightarrow {\mathbb {C}}\) be zonal and positive definite. Under the notation of (14) and (15), we obtain \({\mathcal {H}}_{2\lambda }^{d} \cong {\mathcal {S}}(K^i_\lambda ) \subset H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d})\), for each \(i=1,\ldots ,|{\mathcal {I}}^d_\lambda |\), and the multiplicity of \({\mathcal {H}}_{2\lambda }^{d}\) in \({\mathcal {S}}(K)\) equals the rank of the Fourier coefficient \(\hat{K}_{\lambda }\), implying the orthogonal decomposition

$$\begin{aligned} {\mathcal {S}}(K) = \bigoplus _{\lambda \in \Lambda ^d_{\mathcal {I}}} \bigoplus _{i=1}^{{{\mathrm{rank}}}({\hat{K}}_{\lambda })} {\mathcal {S}}(K^i_\lambda ). \end{aligned}$$
(16)

Proof

Mercer’s theorem implies that the reproducing kernel Hilbert space \({\mathcal {S}}(K)\) decomposes into the pairwise orthogonal eigenspaces with nonzero eigenvalues associated with the integral operator \(T_{K}:L^{2}({\mathcal {G}}_{{\mathcal {I}},d}) \rightarrow L^{2}({\mathcal {G}}_{{\mathcal {I}},d})\) defined by \( T_{K} f(P) := (f,K(P,\cdot ))_{{\mathcal {G}}_{{\mathcal {I}},d}}\), \( P \in {\mathcal {G}}_{{\mathcal {I}},d}\), \(f\in L^{2}({\mathcal {G}}_{{\mathcal {I}},d})\). This decomposition corresponds to the eigenspace decomposition of \(T_{K}\) in the subspace \(H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d})\). More precisely, the convolution property yields that the kernels \(K^i_\lambda \) satisfy, for \(i,j=1,\ldots ,|{\mathcal {I}}^d_\lambda |\),

$$\begin{aligned}&(T_{K} K^i_\lambda (P,\cdot ))(Q) = \alpha _{\lambda }^i K^i_\lambda (P,Q),\\&(K^i_\lambda (P,\cdot ),K^j_\lambda (Q,\cdot ))_{{\mathcal {G}}_{{\mathcal {I}},d}} = \delta _{i,j} K^i_\lambda (P,Q),\\&P,Q \in {\mathcal {G}}_{{\mathcal {I}},d}. \end{aligned}$$

Hence, \(K^i_\lambda \) are the reproducing kernels for the pairwise orthogonal spaces \({\mathcal {S}}(K^i_\lambda )\) with respect to the standard inner product. The convolution property (12) yields that \({{\mathrm{Tr}}}(XP_\lambda (P,\cdot ))\in H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d})\), for any matrix \(X\in {\mathbb {C}}^{|{\mathcal {I}}^d_\lambda |\times |{\mathcal {I}}^d_\lambda |}\). Thus, we infer \({\mathcal {S}}(K^i_\lambda ) \subset H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d})\), so that

$$\begin{aligned} \bigoplus _{i=1}^{|{\mathcal {I}}^d_\lambda |} {\mathcal {S}}(K^i_\lambda ) \subset H_{\lambda }({\mathcal {G}}_{{\mathcal {I}},d}) \cong |{\mathcal {I}}^d_\lambda | {\mathcal {H}}_{2\lambda }^{d},\qquad |\lambda | \ge 0. \end{aligned}$$

Since the spaces \({\mathcal {S}}(K^i_\lambda ) \ne \{0\}\), \(i=1,\ldots ,|{\mathcal {I}}^d_\lambda |\), are orthogonally invariant and pairwise orthogonal, we obtain \({\mathcal {S}}(K^i_\lambda ) \cong {\mathcal {H}}_{2\lambda }^d\), which yields (16). \(\square \)

Let \(K_{2\lambda }:{\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}} \times {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}} \rightarrow {\mathbb {R}}\) denote the reproducing kernel with respect to the differentiation inner product of the irreducible representation \({\mathcal {H}}_{2\lambda }^{d}\) in \({{\mathrm{Hom}}}_{|\lambda |}({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})\). One of the ingredients for the following proof of Theorem 5.1 is that the restriction \(K_{2\lambda }|_{{\mathcal {G}}_{k,d}\times {\mathcal {G}}_{{l},d}}\) coincides with \(p^{k,l}_\lambda \) up to a multiplicative constant. Indeed, we shall follow the strategy in [34]:

Proof of Theorem 5.1

Consider the positive definite zonal kernel

$$\begin{aligned} K:{\mathcal {G}}_{d}\times {\mathcal {G}}_{d} \rightarrow {\mathbb {R}}, \qquad K(P,Q):= C_{\lambda }(P Q),\qquad P,Q \in {\mathcal {G}}_{d}. \end{aligned}$$

The relation (28) implies \({\mathcal {S}}(K)=F_\lambda ({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}})|_{{\mathcal {G}}_{d}}\). Theorem 5.5 and (5) yield

$$\begin{aligned} K = {{\mathrm{Tr}}}({\hat{K}}_{\lambda }^{\top } P_{\lambda }) + \sum _{|\lambda '| < |\lambda |,\;\ell (\lambda ')\le \frac{d}{2}} {{\mathrm{Tr}}}(\hat{K}_{\lambda '}^{\top } P_{\lambda '}). \end{aligned}$$

Furthermore, (5) implies that the irreducible representation \({\mathcal {H}}_{2\lambda }^{d}\) of \({\mathcal {O}}(d)\) occurs exactly once in \({\mathcal {S}}(K)\), so that Theorem 5.5 yields \({\mathcal {S}}(v_{\lambda }^{\top } P_{\lambda } v_{\lambda }) \cong H_{\lambda }^{|\lambda |}({\mathcal {G}}_{{\mathcal {I}},d})\), where \(\hat{K}_{\lambda } = v_{\lambda }^{\top }v_{\lambda }\) for some nonzero vector \(v_{\lambda } \in {\mathbb {R}}^{|{\mathcal {I}}^d_\lambda |}\) with \({\mathcal {I}}=\{1,\ldots ,d-1\}\). According to \(v_{\lambda }^{\top }P_{\lambda } v_{\lambda }={{\mathrm{Tr}}}(\hat{K}_{\lambda }^{\top } P_{\lambda })\), we obtain

$$\begin{aligned} v_{\lambda }^{\top }P_{\lambda }(P,Q) v_{\lambda } = C_{\lambda }(PQ) - \sum _{|\lambda '| < |\lambda |,\;\ell (\lambda ')\le \frac{d}{2}} {{\mathrm{Tr}}}( {\hat{K}}_{\lambda '}^{\top } P_{\lambda '}(P,Q)), \qquad P,Q \in {\mathcal {G}}_{d}.\nonumber \\ \end{aligned}$$
(17)

Note that \(v_{\lambda }^{\top }P_{\lambda } v_{\lambda }\) coincides up to a multiplicative factor with the restriction of the kernel \(K_{2\lambda }\). We shall verify in the following that the kernel \(v_{\lambda }^{\top } P_{\lambda } v_{\lambda }\) reflects the expansion of \(p_{\lambda }\) into zonal polynomials \(C_{\lambda '}\) defined in Theorem 5.1. Starting with (10), we exploit the vanishing and symmetry properties of the Jacobi polynomials, cf. Appendix B.3, combined with the symmetry relations of the intertwining functions, cf. Appendix B.2. We observe that, for any partition \(\lambda \) with \(\ell (\lambda ) \le \tfrac{d}{2}\) and any \((P,Q) \in {\mathcal {G}}_{k,d}\times G_{l,d}\) with \(1\le k,l\le d-1\),

$$\begin{aligned} p_{\lambda }(P,Q)= & {} \sum _{\lambda '\le \lambda } c_{\lambda ,\lambda '}^{\frac{d}{2}} q_{\lambda ,\lambda '}(\tfrac{k}{2}) q_{\lambda ,\lambda '}(\tfrac{l}{2}) C_{\lambda '}(PQ)\nonumber \\= & {} {\left\{ \begin{array}{ll} (b_{\lambda }^{k,l,d})^{-1} p_{\lambda }^{k,l}(P,Q), &{} \ell (\lambda ) \le k,l \le d-\ell (\lambda ),\\ 0, &{} \text {else}, \end{array}\right. } \end{aligned}$$
(18)

where \(b_{\lambda }^{k,l,d} \in {\mathbb {R}}\setminus \{0\}\). After inserting the expansion from (18) into both sides of (17) via \(P_\lambda \) and \(P_{\lambda '}\), we aim to compare coefficients of the zonal polynomials. Let \(k,l \in {\mathcal {I}}=\{\ell (\lambda ),\ldots ,d-\ell (\lambda )\}\) be fixed. One can (only) show linear independence of the functions

$$\begin{aligned} (P,Q) \mapsto C_{\lambda '}(P Q), \quad (P,Q) \in {\mathcal {G}}_{k,d} \times {\mathcal {G}}_{{l},d},\quad \ell (\lambda ')\le \min (k,d-k,l,d-l). \end{aligned}$$

Since \(\lambda ' \le \lambda \) implies \(\ell (\lambda ') \le \min (k,d-k,l,d-l)\), the zonal polynomials in (18) are linearly independent. By applying \(P_{\lambda '}(P,Q) = 0\), \((P,Q) \in {\mathcal {G}}_{k,d} \times {\mathcal {G}}_{{l},d}\), for \(\ell (\lambda ') >\min (k,d-k,l,d-l)\), the summation on the right-hand side in (17) reduces accordingly, and comparing coefficients is justified. Hence, we obtain \(1 = v_{\lambda }^{k}v_{\lambda }^{{l}}\, b_{\lambda }^{k,{l},d} c_{\lambda ,\lambda }^{\frac{d}{2}}\), so that \(c_{\lambda ,\lambda }^{\frac{d}{2}}=1\) leads to

$$\begin{aligned} p_\lambda |_{{\mathcal {G}}_{k,d}\times {\mathcal {G}}_{{l},d}}=v_\lambda ^k v_\lambda ^l p_\lambda ^{k,l},\qquad \ell (\lambda )\le \min (k,d-k,l,d-l). \end{aligned}$$

In other words, we have verified that, for \(\ell (\lambda )\le {{\mathrm{Tr}}}(P), {{\mathrm{Tr}}}(Q) \le d - \ell (\lambda )\),

$$\begin{aligned} v_{\lambda }^{\top }P_{\lambda }(P,Q) v_{\lambda } = \sum _{\lambda ' \le \lambda } c^{\frac{d}{2}}_{\lambda ,\lambda '} q_{\lambda ,\lambda '}(\tfrac{1}{2} {{\mathrm{Tr}}}(P)) q_{\lambda ,\lambda '}(\tfrac{1}{2} {{\mathrm{Tr}}}(Q)) C_{\lambda '}(P Q)=p_\lambda (P,Q).\nonumber \\ \end{aligned}$$
(19)

For the remaining cases, we observe that \(v_{\lambda }^{\top }P_{\lambda }(P,Q) v_{\lambda } = 0\), and \(p_{\lambda }(P,Q) = 0\) holds due to (18). \(\square \)

For \(|\lambda |=0,1,2\), we present the kernels \(K_{2\lambda }\) and \(p_{\lambda }\) as well as \(p^{k,{l}}_{\lambda }\) and \(v_{\lambda }^{k}\) in Appendix D. We now replace \(P_\lambda \) with \(p_\lambda \) and still have a suitable Fourier expansion:

Corollary 5.6

For \({\mathcal {I}} \subset \{1,\ldots ,d -1 \}\), let \(K : {\mathcal {G}}_{{\mathcal {I}},d} \times {\mathcal {G}}_{{\mathcal {I}},d} \rightarrow {\mathbb {C}}\) be a positive definite zonal kernel. Then there is a unique symmetric function \({\widehat{K}}_\lambda :{\mathcal {I}}^d_\lambda \times {\mathcal {I}}^d_\lambda \rightarrow {\mathbb {C}}\) such that

$$\begin{aligned} K(P,Q) = \sum _{\lambda \in \Lambda ^d_{\mathcal {I}}} {\widehat{K}}_\lambda ({{\mathrm{Tr}}}(P),{{\mathrm{Tr}}}(Q)) p_{\lambda }(P,Q), \qquad P,Q \in {\mathcal {G}}_{{\mathcal {I}},d}, \end{aligned}$$

and the multiplicity of \({\mathcal {H}}_{2\lambda }^{d}\) in \(\mathcal S(K)\) is the rank of the matrix \(\big ({\widehat{K}}_\lambda (k,{l})\big )_{k,{l}\in {\mathcal {I}}^d_\lambda }\).

Proof

The Fourier expansion (13) with coefficients \(\hat{K}_{\lambda }^{k,{l}}\) yields \({\widehat{K}}_\lambda (k,{l}) = (v_{\lambda }^{k})^{-1}{\hat{K}}_{\lambda }^{k,{l}} (v_{\lambda }^{{l}})^{-1}\), where \(v_{\lambda }\) is given in (19). Therefore, the rank of \(\big ({\widehat{K}}_\lambda (k,{l})\big )_{k,{l}\in {\mathcal {I}}^d_\lambda }\) is the same as the one of the original Fourier coefficient, so that Theorem 5.5 implies the statement. \(\square \)

The following result is an important step forward:

Theorem 5.7

For \({\mathcal {I}}=\{1,\ldots ,d-1\}\) and any partition \(\lambda \) with \(\ell (\lambda ) \le \tfrac{d}{2}\) and \(s=0,\ldots ,|{\mathcal {I}}^d_\lambda |-1\), the multiplicity of \({\mathcal {H}}^d_{2\lambda }\) in \(H_{\lambda }^{|\lambda |+s}({\mathcal {G}}_{d})\) is \(s+1\).

Proof

For any partition \(\lambda \) with \(\ell (\lambda ) \le \tfrac{d}{2}\), and \(i=0,\ldots ,|{\mathcal {I}}^d_\lambda |-1\), there are polynomials \(q_{\lambda }^{i}:{\mathbb {R}}\rightarrow {\mathbb {R}}\) of degree i satisfying the orthogonality relations

$$\begin{aligned} \sum _{m=\ell (\lambda )}^{d-\ell (\lambda )} q_{\lambda }^{i}(m)q_{\lambda }^{j}(m) |v_{\lambda }^{m}|^{2} = \delta _{i,j}\qquad i,j=0,\ldots ,|{\mathcal {I}}^d_\lambda |-1, \end{aligned}$$

where \(v_{\lambda }\) is given in (19). We can define the associated positive definite kernels \(K_{\lambda }^{i}:{\mathcal {G}}_{d}\times {\mathcal {G}}_{d}\rightarrow {\mathbb {R}}\) by

$$\begin{aligned} K_{\lambda }^{i}(P,Q) := q_{\lambda }^{i}(P)q_{\lambda }^{i}(Q) p_{\lambda }(P,Q), \qquad P,Q \in {\mathcal {G}}_{d}. \end{aligned}$$
(20)

For \(P \in {\mathcal {G}}_{k,d}\), \(Q\in {\mathcal {G}}_{{l},d}\), the identity

$$\begin{aligned} \int _{{\mathcal {G}}_{m,d}}p_{\lambda }(P,R)p_{\lambda }(R,Q)\mathrm {d}\sigma _{m,d}(R)&= v_{\lambda }^{k} |v_{\lambda }^{m}|^{2} v_{\lambda }^{{l}} \int _{{\mathcal {G}}_{m,d}}p_{\lambda }^{k,m}(P,R)p_{\lambda }^{m,{l}}(R,Q) \mathrm {d}\sigma _{m,d}(R) \\&= |v_{\lambda }^{m}|^{2} p_{\lambda }(P,Q), \end{aligned}$$

implies \(K_{\lambda }^{i} * K_{\lambda }^{j} = \delta _{i,j} K_{\lambda }^{j}\). Hence, Corollary 5.6 leads to

$$\begin{aligned} \bigoplus _{i=0}^{s} {\mathcal {S}}(K^{i}_\lambda ) \subset H_{\lambda }^{|\lambda |+s}({\mathcal {G}}_{d}),\qquad s=0,\ldots ,|{\mathcal {I}}^d_\lambda |-1. \end{aligned}$$
(21)

Note that (21) also implies the lower bound on the multiplicities in Corollary 3.2. We shall complete the proof in Appendix C, where we verify that (21) holds with equality; i.e., we decompose \(H_{\lambda }^{|\lambda |+s}({\mathcal {G}}_{d})\) into \(s+1\) orthogonal subspaces of increasing polynomial degree with simple multiplicities. \(\square \)

We are now able to determine the multiplicities of the irreducible components in \({{\mathrm{Pol}}}_t({\mathcal {G}}_{{\mathcal {I}},d})\).

Theorem 5.8

For \({\mathcal {I}} \subset \{1,\ldots , d-1\}\) and \(t\ge 0\), the following holds:

$$\begin{aligned} \mu ^d_{\lambda }({\mathcal {I}},t) = {\left\{ \begin{array}{ll} \min \{ t - |\lambda |+1,|{\mathcal {I}}^d_\lambda |\}, &{} 0\le |\lambda |\le t, \quad \lambda \in \Lambda ^d_{\mathcal {I}}, \\ 0, &{} \text {else}. \end{array}\right. } \end{aligned}$$

Proof

According to Corollary 3.2, we only need to suitably bound \( \mu ^d_{\lambda }({\mathcal {I}},t)\) from above. Proposition 5.7 yields \(\mu _{\lambda }^{d}(\{1,\ldots ,d-1\}^d_\lambda ,t ) \le t -|\lambda |+1\). Since \(\mu _{\lambda }^{d}({\mathcal {I}},t) \le \mu _{\lambda }^{d}(\{1,\ldots ,d-1\}^d_\lambda ,t )\) and the general upper bound \(\mu _{\lambda }^{d}({\mathcal {I}},t) \le |{\mathcal {I}}^d_\lambda |\) holds, we conclude the proof. \(\square \)

The knowledge of \(\dim ({\mathcal {H}}^d_{2\lambda })\) through [27, Formulas (24.29) and (24.41)] enables us to compute the dimension of \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\). Moreover, counting irreducible components yields

$$\begin{aligned} {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d}) \cong \bigoplus _{i=1}^{s}{{\mathrm{Pol}}}_{t-i+1}({\mathcal {G}}_{k_{i},d}),\quad s:=\min \{t+1,|{\mathcal {I}}|\}, \end{aligned}$$
(22)

where \({\mathcal {I}} =\{k_{i}\}_{i=1}^r\) is ordered such that \(\min \{k_{1},d-k_{1}\} \ge \cdots \ge \min \{k_{r},d-k_{r}\}\), so that equality actually holds in Theorem 3.1. Our proof of Theorem 3.1 then reveals the intriguing identity

$$\begin{aligned} {{\mathrm{null}}}(|_{{\mathcal {G}}_{k_1,d}}) = ({{\mathrm{Tr}}}(\cdot ) - k_1) {{\mathrm{Pol}}}_{t-1}({\mathcal {G}}_{{\mathcal {I}},d}),\qquad t\ge 1, \end{aligned}$$
(23)

with the restriction mapping \(|_{{\mathcal {G}}_{k_1,d}}: {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\rightarrow {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{k_1,d})\).

6 Cubatures and Designs on Unions of Grassmannians

So far, we have analyzed the irreducible decomposition of polynomial spaces on unions of Grassmannians. Our results enable us in the following to study cubatures on unions of Grassmannians.

6.1 Introducing Cubatures and Designs

Any orthogonally invariant finite signed measure \(\sigma _{{\mathcal {I}},d}\) on \({\mathcal {G}}_{{\mathcal {I}},d}\) is a linear combination of the Haar (probability) measures \(\sigma _{k,d}\), \(k \in {\mathcal {I}}\), i.e.,

$$\begin{aligned} \sigma _{{\mathcal {I}},d} = \sum _{k\in {\mathcal {I}}} m_k \sigma _{k,d}, \end{aligned}$$

for some \(\{m_k\}_{k\in {\mathcal {I}}}\subset {\mathbb {R}}\). For points \(\{P_j\}_{j=1}^n\subset {\mathcal {G}}_{{\mathcal {I}},d}\) and weights \(\{\omega _j\}_{j=1}^n \subset {\mathbb {R}}\), we say that \(\{(P_j,\omega _j)\}_{j=1}^n\) is a cubature for \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) (resp. \({{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\)) with respect to \(\sigma _{{\mathcal {I}},d}\) if

$$\begin{aligned} \int _{{\mathcal {G}}_{{\mathcal {I}},d}} f(P)d\sigma _{{\mathcal {I}},d}(P) = \sum _{j=1}^n \omega _j f(P_j),\qquad f\in {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\quad (\mathrm {resp.}\; f\in {{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})). \end{aligned}$$

If \(\{(P_j,\omega _j)\}_{j=1}^n\) is a cubature for \({{\mathrm{Hom}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) with \(\sum _{k \in {\mathcal {I}}} m_{k} = \sum _{j=1}^{n} \omega _{j}\), then it is also a cubature for \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\), cf. Theorem 4.3. The value of the parameter t is often called the strength of the cubature.

Remark 6.1

According to [19, Propositions 2.6 and 2.7], there is a cubature \(\{(P_j,\omega _j)\}_{j=1}^n\) with nonnegative weights \(\{\omega _j\}_{j=1}^n\) for \({{\mathrm{Pol}}}_{ t}({\mathcal {G}}_{{\mathcal {I}},d})\) with respect to \(\sigma _{{\mathcal {I}},d}\) such that \(n\le \dim ({{\mathrm{Pol}}}_{ t}({\mathcal {G}}_{{\mathcal {I}},d}))\). For \(m_k\ne 0\), \(k\in {\mathcal {I}}\), any cubature \(\{(P_j,\omega _j)\}_{j=1}^n\) for \({{\mathrm{Pol}}}_{2t}({\mathcal {G}}_{{\mathcal {I}},d})\) with respect to \(\sigma _{{\mathcal {I}},d}\) with nonnegative weights satisfies \(n\ge \dim ( {{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d}))\), cf. [19, Proposition 1.7].

Analogously to Euclidean designs, cf. [7,8,9,10], cubatures on unions of Grassmannians induce cubatures on single Grassmannians, but potentially with lower strength:

Proposition 6.2

If \(\{(P_j,\omega _j)\}_{j=1}^n\) is a cubature for \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\), \(t\ge |{\mathcal {I}}|-1\), with respect to the signed measure \(\sigma _{{\mathcal {I}},d}=\sum _{k\in {\mathcal {I}}} m_k \sigma _{k,d}\), then, for any \(k\in {\mathcal {I}}\),

$$\begin{aligned} \{ (P_j,\omega _j) : P_j\in {\mathcal {G}}_{k,d},\; j=1,\ldots ,n\} \end{aligned}$$

is a cubature for \({{\mathrm{Pol}}}_{s}({\mathcal {G}}_{k,d})\) with respect to the signed measure \(m_{k} \sigma _{k,d}\), where \(s=t-|{\mathcal {I}}|+1\).

Proof

Let f be a polynomial of degree at most s on \({\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\); then we know that \(1_{{\mathcal {G}}_{k,d}} \!\cdot f\) is a polynomial of degree at most t. Hence, the statement follows from

$$\begin{aligned} m_{k} \int _{{\mathcal {G}}_{k,d}} f(P)d\sigma _{k,d} = \int _{{\mathcal {G}}_{{\mathcal {I}},d}} (1_{{\mathcal {G}}_{k,d}}\! \cdot f)(P)d\sigma _{{\mathcal {I}},d} = \sum _{\{j:P_j\in {\mathcal {G}}_{k,d}\}} \omega _j f(P_j). \end{aligned}$$

\(\square \)

We also observe that any cubature of strength 2t gives rise to a cubature of strength \(2t+1\):

Proposition 6.3

Let \(\{(P_j,\omega _j)\}_{j=1}^n\) be a cubature for \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) with respect to the signed measure \(\sigma _{{\mathcal {I}},d}=\sum _{k\in {\mathcal {I}}} m_k \sigma _{k,d}\). Then

$$\begin{aligned} \{(P_j,\omega _j)\}_{j=1}^n \cup \{(I-P_j,(-1)^t\omega _j)\}_{j=1}^n \end{aligned}$$

is a cubature for \({{\mathrm{Pol}}}_{t+1}({\mathcal {G}}_{{\mathcal {I}}\cup (d-{\mathcal {I}}),d})\) with respect to the signed measure

$$\begin{aligned} \sigma _{{\mathcal {I}}\cup (d-{\mathcal {I}}),d}=\sum _{k\in {\mathcal {I}}} m_k(\sigma _{k,d} +(-1)^t\sigma _{d-k,d}). \end{aligned}$$

Proof

In view of Remark 4.4, it is sufficient to consider the polynomial \({{\mathrm{Tr}}}( X \cdot )^{t+1}|_{{\mathcal {G}}_{{\mathcal {I}}\cup (d-{\mathcal {I}}),d}}\) for \(X\in {\mathbb {R}}^{d\times d}_{{{\mathrm{sym}}}}\). Then we have

$$\begin{aligned} \int _{{\mathcal {G}}_{{\mathcal {I}}\cup (d-{\mathcal {I}}),d} }\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {{\mathrm{Tr}}}( XP)^{t+1} d\sigma _{{\mathcal {I}}\cup (d-{\mathcal {I}}),d}(P)= & {} \int _{{\mathcal {G}}_{{\mathcal {I}},d}} \!\!\!\!\!\!{{\mathrm{Tr}}}( XP)^{t+1}d\sigma _{{\mathcal {I}},d} (P)\\&+ \!\! \int _{{\mathcal {G}}_{{\mathcal {I}},d}} \!\!\!\!\!\!{{\mathrm{Tr}}}( X(I-P))^{t+1}(-1)^td\sigma _{{\mathcal {I}},d} (P) \\= & {} \int _{{\mathcal {G}}_{{\mathcal {I}},d}} \!\!\!\!\!\!\big ({{\mathrm{Tr}}}(XP)^{t+1} + (-1)^t{{\mathrm{Tr}}}(X(I-P))^{t+1} \big )d\sigma _{{\mathcal {I}},d}(P). \end{aligned}$$

The mapping \(P\mapsto {{\mathrm{Tr}}}( XP)^{t+1} + (-1)^t{{\mathrm{Tr}}}( X(I-P))^{t+1} \) is contained in \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\), because the two terms with exponent \(t+1\) cancel out. Thus, \(\{(P_j,\omega _j)\}_{j=1}^n\) being a cubature for \({{\mathrm{Pol}}}_{t}({\mathcal {G}}_{{\mathcal {I}},d})\) yields

$$\begin{aligned}&\int _{{\mathcal {G}}_{{\mathcal {I}}\cup (d-{\mathcal {I}}),d} }\!\!\!\!\!\! {{\mathrm{Tr}}}( XP)^{t+1} d\sigma _{{\mathcal {I}}\cup (d-{\mathcal {I}}),d}(P)\\&\quad =\sum _{j=1}^n \omega _j \big ( {{\mathrm{Tr}}}(XP_j)^{t+1} + (-1)^t{{\mathrm{Tr}}}( X(I-P_j))^{t+1} \big ). \end{aligned}$$

\(\square \)

Cubatures of strength t on single Grassmannians, whose weights are all the same, are called t-designs and have been studied in [2,3,4,5]. We shall extend this concept to unions of Grassmannians after a brief observation on characterstic functions. For \(t=|{\mathcal {I}}|-1\) with \({\mathcal {I}}=\{k_0,\ldots ,k_{t}\}\), the Vandermonde matrix \(V=(k_i^j)_{i,j=0,\ldots ,t}\in {\mathbb {R}}^{t+1}\) is invertible. Hence, for \(i=0,\ldots ,t\), there are \(\alpha _i:=(\alpha _{i,0},\ldots ,\alpha _{i,t})^\top \in {\mathbb {R}}^{t+1}\) with \(V \alpha _i=e_{i+1}\) implying

$$\begin{aligned} \sum _{j=0}^{t} \alpha _{i,j} {{\mathrm{Tr}}}(P)^j={\left\{ \begin{array}{ll} 1,&{} P\in {\mathcal {G}}_{k_i,d},\\ 0,&{} P\in {\mathcal {G}}_{{\mathcal {I}},d}\setminus {\mathcal {G}}_{k_i,d}. \end{array}\right. } \end{aligned}$$

Thus, the characteristic function of each \({\mathcal {G}}_{k,d}\) is contained in \({{\mathrm{Pol}}}_t({\mathcal {G}}_{{\mathcal {I}},d})\), for all \(t\ge |{\mathcal {I}}|-1\). The latter yields that any cubature \(\{(P_j,\omega _j)\}_{j=1}^n\) for \({{\mathrm{Pol}}}_t({\mathcal {G}}_{{\mathcal {I}},d})\) with respect to \(\sigma _{{\mathcal {I}},d}=\sum _{k\in {\mathcal {I}}}m_k \sigma _{k,d}\) satisfies \( m_k=\sum _{\{j:P_j\in {\mathcal {G}}_{k,d}\}} \omega _j\), for \(k\in {\mathcal {I}}\). If weights are the same on each single \({\mathcal {G}}_{k,d}\), then those weights must be \(\frac{m_k}{n_k}\), where \(n_k=|\{j:P_j\in {\mathcal {G}}_{k,d}\}|\). Indeed, we impose this condition in our definition of designs:

Definition 6.4

Let \({\mathcal {P}}_k\subset {\mathcal {G}}_{k,d}\) be finite, and set \(n_k:=|{\mathcal {P}}_k|\). The collection \({\mathcal {P}}_{\mathcal {I}}=\bigcup _{k\in {\mathcal {I}}}{\mathcal {P}}_k\) is called a t-design with respect to an orthogonally invariant signed measure \(\sigma _{{\mathcal {I}},d}=\sum _{k\in {\mathcal {I}}}m_k \sigma _{k,d}\) if

$$\begin{aligned} \int _{{\mathcal {G}}_{{\mathcal {I}},d}} f(P)d\sigma _{{\mathcal {I}},d}(P) = \sum _{k\in {\mathcal {I}}} \tfrac{m_k}{n_k} \sum _{P\in {\mathcal {P}}_k} f(P) \quad \text {for all } f\in {{\mathrm{Pol}}}_t({\mathcal {G}}_{{\mathcal {I}},d}). \end{aligned}$$

Thus, any t-design is a cubature, whose weights are the same on each Grassmannian but can differ across different Grassmannians. For \(|{\mathcal {I}}|=1\), our definition reduces to the standard Grassmannian designs as considered in [2,3,4,5]. The existence of t-designs with \(t=1\) in single Grassmannians was studied in [14]. For a discussion on the existence of 1-designs in unions of Grassmannians with \(\frac{m_k}{n_k}=\frac{m_{l}}{n_{l}}\), for all \(k,{l}\in {\mathcal {I}}\), we refer to [12].

6.2 Constructing Cubatures and Designs by Numerical Minimization

To construct cubatures or designs, we consider the t-fusion frame potential, cf. [6],

$$\begin{aligned} {{\mathrm{FFP}}}_t(\{(P_j,\omega _j)\}_{j=1}^n):=\sum _{i,j=1}^{n}\omega _i\omega _j{{\mathrm{Tr}}}(P_iP_j)^t, \end{aligned}$$

where \(\{P_j\}_{j=1}^n\subset {\mathcal {G}}_{{\mathcal {I}},d}\) and \(\{\omega _j\}_{j=1}^n\). The 1-fusion frame potential was already investigated in [13, 38]. Lower bounds on the t-fusion frame potential for general positive integers t were derived in [6]. Only for single Grassmannians, i.e., \(|{\mathcal {I}}|=1\), those \(\{(P_j,\omega _j)\}_{j=1}^n\) were characterized in [6], for which the bounds are matched. In the following, we provide a characterization for the general case \(|{\mathcal {I}}| \ge 1\). Before we state this result though, it is convenient to define

$$\begin{aligned} {\mathcal {T}}_{\sigma _{{\mathcal {I}},d}}(t):=\int _{{\mathcal {G}}_{{\mathcal {I}},d}}\int _{{\mathcal {G}}_{{\mathcal {I}},d}} {{\mathrm{Tr}}}( PQ)^td\sigma _{{\mathcal {I}},d}(P)d\sigma _{{\mathcal {I}},d}(Q) = m^{\top } T_{{\mathcal {I}},d}(t) m, \end{aligned}$$
(24)

where \(\sigma _{{\mathcal {I}},d}=\sum _{k\in {\mathcal {I}}}m_k \sigma _{k,d}\) and \(m=(m_{k})_{k\in {\mathcal {I}}}\) with the matrix \(T_{{\mathcal {I}},d}(t) \in {\mathbb {R}}^{|{\mathcal {I}}|\times |{\mathcal {I}}|}\) being given by

$$\begin{aligned} (T_{{\mathcal {I}},d}(t))_{k,{l}}:= \int _{{\mathcal {G}}_{{l},d}}\int _{{\mathcal {G}}_{k,d}}{{\mathrm{Tr}}}(P Q)^{t} d\sigma _{k,d}(P) d\sigma _{{l},d}(Q). \end{aligned}$$

Note that \(T_{{\mathcal {I}},d}(t)\) is the (0)-th Fourier coefficient of the positive definite zonal kernel \(K_{t}(P,Q) = {{\mathrm{Tr}}}( P Q)^{t}\), for \(P,Q \in {\mathcal {G}}_{{\mathcal {I}},d}\), and thus symmetric and positive semidefinite.

Theorem 6.5

Given \(\{P_j\}_{j=1}^n\subset {\mathcal {G}}_{{\mathcal {I}},d}\) with weights \(\{\omega _j\}_{j=1}^n \subset {\mathbb {R}}\), let \(m_{k}=\sum _{\{j:P_j\in {\mathcal {G}}_{k,d}\}}\omega _j\). Then the fusion frame potential is bounded from below by

$$\begin{aligned} {{\mathrm{FFP}}}_t(\{(P_j,\omega _j)\}_{j=1}^n)\ge {\mathcal {T}}_{\sigma _{{\mathcal {I}},d}}(t). \end{aligned}$$
(25)

Equality holds if and only if \(\{(P_j,\omega _j)\}_{j=1}^n\) is a cubature for \({{\mathrm{Pol}}}_{ t}({\mathcal {G}}_{{\mathcal {I}},d})\) with respect to \(\sigma _{{\mathcal {I}},d}\).

This theorem is an extension of results in [6], where the lower bound is already derived but equality is not discussed. For the proof, we refer to Appendix A.2.

Theorem 6.5 enables the use of numerical minimization schemes to derive cubatures. Knowledge of the global lower bound \({\mathcal {T}}_{\sigma _{{\mathcal {I}},d}}(t)\) is important to check if numerical solutions are indeed cubatures by ruling out that the minimization got stuck in a local minimum. The matrix \(T_{{\mathcal {I}},d}(t)\) can be computed via zonal polynomials by

$$\begin{aligned} T_{{\mathcal {I}},d}(t) = \sum _{\begin{array}{c} |\lambda | = t,\\ \ell (\lambda ) \le d/2 \end{array} } \frac{c_{\lambda } c_{\lambda }^{\top }}{C_{\lambda }(I_{d})} \in {\mathbb {R}}^{|{\mathcal {I}}| \times |{\mathcal {I}}|}, \qquad c_{\lambda }:=(C_{\lambda }(I_{k}))_{k \in {\mathcal {I}}} \in {\mathbb {R}}^{|{\mathcal {I}}|}, \end{aligned}$$

cf. (30) in Appendix B.1, and note that \(C_{\lambda }(I_k)\) is explicitly computed in [15, 30, 39].

For suitable minimization algorithms on Grassmannians, we refer to [1, 11, 29]. In Sect. 7 we shall indeed numerically minimize the fusion frame potential with equal weights and check that the lower bound is attained.

7 Examples of t-Designs Derived from Numerical Minimization

Here, we shall construct some families of 1-, 2-, and 3-designs in unions of Grassmannians. By numerically minimizing the energy functional (25) using a conjugate gradient approach, cf. [29, Section 3.3.1], we compute candidates for t-designs, i.e., t-designs up to machine precision. Based on the special structures of the Gram matrices of these numerical minimizers, we looked for group orbits that describe them analytically. Indeed “beautifying” our numerical results, we were able to analytically specify our candidates, which turned out to be exact minimizers.

7.1 A Family of 1-Designs in Arbitrary Dimensions

We analytically construct d lines and 1 hyperplane in \({\mathbb {R}}^d\), so that the corresponding orthogonal projectors are a 1-design in \({\mathcal {G}}_{1,d}\cup {\mathcal {G}}_{d-1,d}\) with respect to the measure \(\sigma _{\{1,d-1\},d}=m_{1}\sigma _{1,d}+m_{d-1}\sigma _{d-1,d}\), with \(m_{1}=1\) and \(m_{d-1}\in [-1/(d-1),1]\). Indeed, let the d lines be described by the vectors

$$\begin{aligned} a_{i} := \sqrt{1-m_{d-1}}\, e_{i} + \frac{1}{d}\big (\sqrt{1+(d-1)m_{d-1}}-\sqrt{1-m_{d-1}} \big ) e\, \in {\mathbb {R}}^{d}, \qquad i=1,\ldots ,d, \end{aligned}$$

where \(e_{i} \in {\mathbb {R}}^{d}\) are the standard unit vectors and \(e:=e_{1}+\cdots +e_{d}\). The associated rank-1 projectors are \(P_{i}:=a_{i}\, a_{i}^{\top } \in {\mathbb {R}}^{d}\), \(i=1,\ldots ,d,\) and the rank-\((d-1)\) projector is \(P_{d+1}:=I_{d} - \frac{1}{d} e\, e^{\top }\). We calculate

$$\begin{aligned} {\mathcal {T}}_{\sigma _{1,d}+m_{d-1}\sigma _{{d-1,d}}}(1) = \tfrac{1}{d}\big ( 1 + (d-1) m_{d-1}\big )^{2} \end{aligned}$$

and observe equality with the 1-fusion frame potential \({{\mathrm{FFP}}}_1\big (\{(P_{j},\omega _{j})\}_{j=1}^{d+1}\big )\) with weights \(\omega _{1}=\cdots =\omega _{d}=\frac{m_{1}}{d}\), \(\omega _{d+1}=m_{d-1}\). According to Theorem 6.5, \(\{(P_{j},\omega _{j})\}_{j=1}^{d+1}\) is a 1-design.

For \(m_{d-1}=1\), the d lines coincide and are orthogonal to the hyperplane. This configuration can also be considered as a single line with one hyperplane forming a 1-design. If \(-1/(d-1)\le m_{d-1}<1\), then this one line is split into d lines forming a \((d-1)\)-dimensional simplex, and, for \(m_{d-1}=-1/(d-1)\), the lines lie in the hyperplane.

Remark 7.1

According to Proposition 6.2, any cubature of strength 1 in \({\mathcal {G}}_{1,d}\cup {\mathcal {G}}_{d-1,d}\) yields parts in \({\mathcal {G}}_{1,d}\) and in \({\mathcal {G}}_{d-1,d}\) that are cubatures of strength 0. Hence, there must be at least one element in each. For \(m_{1}=m_{d-1}=1\), the above example matches this lower bound.

7.2 A Family of 1-Designs in \({\mathbb {R}}^4\)

We construct a family of 2 lines and 2 planes in \({\mathbb {R}}^4\) forming a 1-design with respect to \(m_1\in [-2,2]\) and \(m_2=1\). The rank-1 projectors are \( P_{1}:=e_{1}\,e_{1}^{\top }\), \(P_{2}:=e_{3}\,e_{3}^{\top } \in {\mathbb {R}}^{4}\), and the 2-dimensional projectors are

$$\begin{aligned} P_{3} := \begin{pmatrix} a_{3}\,a_{3}^{\top } &{} 0 \\ 0 &{} a_{3}\,a_{3}^{\top } \end{pmatrix}, \; P_{4} := \begin{pmatrix} a_{4}\,a_{4}^{\top } &{} 0 \\ 0 &{} a_{4}\,a_{4}^{\top } \end{pmatrix} \in {\mathbb {R}}^{4}, \quad a_{3/4}:=\frac{1}{2} \begin{pmatrix} \sqrt{2 - m_{1}}\\ \pm \sqrt{2 + m_{1}} \end{pmatrix} \in {\mathbb {R}}^{2}. \end{aligned}$$

For \(m_{1} \in [-2,2]\), this family provides 1-designs since equality holds in Theorem 6.5, where \( {\mathcal {T}}_{m_{1} \sigma _{1,4}+ \sigma _{2,4}}(1)=\left( 1 + \tfrac{m_{1}}{2} \right) ^{2}. \) For \(m_1=2\), the two planes coincide and the two lines are orthogonal to each other and to the planes. The two planes also coincide for \(m_1=-2\), and then the two lines span the same plane. The choice \(m_1=0\) yields two planes that are orthogonal to each other.

Remark 7.2

For \(m_1,m_2>0\), there do not exist any 1-designs in \({\mathcal {G}}_{1,4}\cup {\mathcal {G}}_{2,4}\) of cardinality 2. In this sense, our example with \(m_1=2\) and \(m_2=1\) is optimal.

7.3 A Family of 2-Designs in \({\mathbb {R}}^3\)

The numerical minimization enabled us to identify an analytic family of 6 lines and 4 planes in \({\mathbb {R}}^3\) forming a 2-design with \(m_1=1\) and \(m_2\in [-3/8,3/2]\). The tetrahedral group \(\mathrm T\) is generated by a cyclic coordinate shift \(R_{1}=e_2e_1^\top +e_3e_2^\top +e_1e_3^\top \in {\mathbb {R}}^{3\times 3}\) and a reflection \(R_{2}=I_3-2e_1e_1^\top \). The 6 lines and 4 planes are obtained as distinct orbits under the canonical action of the tetrahedral group. The rank-1 and rank-2 projectors are

$$\begin{aligned} \begin{aligned} {\mathcal {P}}_{1}&:= \{ R P_{1} R^{\top } \;:\; R \in \mathrm T \}, \qquad P_{1}:=\frac{1}{2} a_{1}\,a_{1}^{\top }, \quad a_{1}:=\Big (\sqrt{1-v},\sqrt{1+v},0\Big )^{\top },\\ {\mathcal {P}}_{2}&:= \{ R P_{2} R^{\top } \;:\; R \in \mathrm T \}, \qquad P_{2}:=I - \frac{1}{3} e\,e^{\top } \in {\mathbb {R}}^{3}, \end{aligned} \end{aligned}$$

respectively, where \(v:=\sqrt{\frac{1}{15}(3+8m_{2})}\). For \(m_{2}\in [-3/8,3/2]\), the union \({\mathcal {P}}={\mathcal {P}}_{1} \cup {\mathcal {P}}_{2}\) is a 2-design since equality holds in Theorem 6.5 for \(t=2\) with \( {\mathcal {T}}_{\sigma _{1,3}+ m_{2}\sigma _{2,3}}(2)=\tfrac{1}{15}\left( 3+16m_{2}+28m_{2}^{2}\right) . \) For \(m_2=3/2\), there are 4 planes, and the 6 lines collapse to the 3 coordinate axes. If \(m_2=-3/8\), then the 6 lines are exactly the 6 intersection lines of the 4 planes. For \(m_2=0\), the 6 lines correspond to the vertices of the icosahedron.

Remark 7.3

Any cubature of strength 2 with nonnegative weights needs at least n distinct cubature points with \(n\ge \dim ({{\mathrm{Pol}}}_1({\mathcal {G}}_{{\mathcal {I}},d}))\), cf. Remark 6.1. Theorem 5.8 leads to \(\dim ({{\mathrm{Pol}}}_1({\mathcal {G}}_{1,3} \cup {\mathcal {G}}_{2,3})) = 7\), so that our construction of 4 planes and 3 lines associated with the choice \(m_1=1\) and \(m_2=3/2\) has the minimal number of elements. Moreover, Proposition 6.2 implies that any 2-design on \({\mathcal {G}}_{1,3} \cup {\mathcal {G}}_{2,3}\) has at least 3 lines.

7.4 A Family of 2-Designs in \({\mathbb {R}}^4\)

We shall provide a family of 8 lines and 8 planes in \({\mathbb {R}}^4\) forming a 2-design for \(m_1=1\) and \(m_2\in [3/4,3/2]\). The lines are generated by a symmetry group \(\mathrm G_{1}\) of cardinality \(|\mathrm G_{1}|= 64\), which is generated by a cyclic coordinate shift \(R_{1}=e_2e_1^\top +e_3e_2^\top +e_4e_3^\top +e_1e_4^\top \in {\mathbb {R}}^{4\times 4}\) and reflection \(R_{2}=I_4-2e_1e_1^\top \). The rank-1 projectors are the orbit

$$\begin{aligned} {\mathcal {P}}_{1}:= \{ R P_{1} R^{\top } \;:\; R \in \mathrm G_{1} \}, \quad P_{1}:=\frac{1}{2} a_{1}\,a_{1}^{\top }, \;\; a_{1}:=\Big (\sqrt{1-v},\sqrt{1+v},0,0\Big )^{\top }, \end{aligned}$$

where \(v:=\sqrt{\frac{1}{9}(8m_{2}-3)}\). The planes are derived from two orbits under a cyclic group \(\mathrm G_{2}\) of cardinality \(|\mathrm G_{2}|=4\) generated by \(R_3=-e_3e_1^\top +e_2e_2^\top +e_1e_3^\top -e_4e_4^\top \in {\mathbb {R}}^{4\times 4}\). More precisely, the set of rank-2 projectors is \({\mathcal {P}}_{2} := \{ R P_{i} R^{\top } \;:\; R \in \mathrm G_{2},\; i=2,3 \}\) with

$$\begin{aligned}\begin{aligned} P_{2/3}&:= \frac{1}{6} \begin{pmatrix} 3\mp \sqrt{1-b^{2}} &{} \sqrt{2-3b+b^{2}} &{} \pm \sqrt{4-b^{2}} &{} \mp \sqrt{2+3b+b^{2}} \\ \sqrt{2-3b+b^{2}} &{} 3\pm \sqrt{1-b^{2}} &{} \sqrt{2+3b+b^{2}} &{} \sqrt{4-b^{2}} \\ \pm \sqrt{4-b^{2}} &{} \sqrt{2+3b+b^{2}} &{} 3\mp \sqrt{1-b^{2}} &{} \mp \sqrt{2-3b+b^{2}} \\ \mp \sqrt{2+3b+b^{2}} &{} \sqrt{4-b^{2}} &{} \mp \sqrt{2-3b+b^{2}} &{} 3\pm \sqrt{1-b^{2}} \end{pmatrix}, \end{aligned} \end{aligned}$$

where \(b:=\sqrt{2-\frac{3}{2m_{2}}}\). For \({\mathcal {P}}:=\mathcal P_{1}\cup {\mathcal {P}}_{2}\), the lower bound \( \mathcal T_{\sigma _{1,4}+m_{2}\sigma _{2,4}}(2)=\tfrac{1}{72} \left( 9 + 48 m_{2} + 80 m_{2}^2\right) \) on the 2-fusion frame potential is matched for \(m_{2}\in [3/4,3/2]\). Therefore, \({\mathcal {P}}\) is indeed a 2-design. For \(m_2=3/2\), each plane intersects two other planes in one line, and the 8 lines coincide in pairs with the 4 coordinate axes.

Remark 7.4

Proposition 6.2 yields that any 2-designs with \(d=4\), \(m_1=1\), and \(m_2=3/2\) must have at least 4 lines, which is matched by our example of 4 lines and 8 planes.

7.5 A 2-Design in \({\mathbb {R}}^5\)

The following collection of 5 lines and 16 planes in \({\mathbb {R}}^5\) forms a 2-design for \(m_1=1\) and \(m_2=5/3\). The lines are given by the 5 coordinate axes, i.e., \({\mathcal {P}}_{1}:=\{ P_{i}: i=1,\ldots ,5 \}\), \(P_{i}:= e_{i} e_{i}^{\top } \in {\mathbb {R}}^{5\times 5}\), for \(i=1,\ldots ,5\). To construct the planes, we consider the transformation group \(\mathrm G\) of cardinality \(|\mathrm G| = 16\), generated by \(G_1=I_3 - e_5e_4^\top +e_4e_5^\top \in {\mathbb {R}}^{5\times 5}\) and \(G_2=e_2e_1^\top -e_1e_2^\top +e_3e_3^\top +e_4e_4^\top -e_5e_5^\top \). The set of rank-2 projectors is the orbit \({\mathcal {P}}_{2} :=\{ R P_{6} R^{\top } \;:\; R \in \mathrm G \}\), where

$$\begin{aligned} P_{6} := \frac{1}{5} \left( {\begin{matrix} 2 &{}\sqrt{\frac{3}{2}} &{}\sqrt{\frac{1}{6}\left( 9 + 5\sqrt{3} \right) } &{}\frac{1}{6}\left( 3 - 5\sqrt{3} \right) &{}\sqrt{\frac{2}{3}} \\ \sqrt{\frac{3}{2}} &{} 2 &{}\sqrt{\frac{1}{6}\left( 9 - 5\sqrt{3} \right) } &{}\sqrt{\frac{2}{3}} &{}\frac{1}{6}\left( 3 + 5\sqrt{3} \right) \\ \sqrt{\frac{1}{6}\left( 9 + 5\sqrt{3} \right) } &{}\sqrt{\frac{1}{6}\left( 9 - 5\sqrt{3} \right) } &{} 2 &{} - \sqrt{\frac{1}{6} \left( 9 + 5\sqrt{3} \right) } &{} - \sqrt{\frac{1}{6}\left( 9 - 5\sqrt{3} \right) } \\ \frac{1}{6}\left( 3 - 5\sqrt{3} \right) &{}\sqrt{\frac{2}{3}} &{} - \sqrt{\frac{1}{6}\left( 9 + 5\sqrt{3} \right) } &{} 2 &{}\sqrt{\frac{3}{2}} \\ \sqrt{\frac{2}{3}} &{}\frac{1}{6}\left( 3 + 5\sqrt{3} \right) &{} - \sqrt{\frac{1}{6}\left( 9 - 5\sqrt{3} \right) } &{}\sqrt{\frac{3}{2}} &{} 2 \end{matrix}} \right) . \end{aligned}$$

For \({\mathcal {P}}:={\mathcal {P}}_{1}\cup {\mathcal {P}}_{2}\), the lower bound \( {\mathcal {T}}_{\sigma _{1,5}+\frac{5}{3} \sigma _{2,5}}(2)=\tfrac{131}{45} \) on the 2-fusion frame potential is matched, so that \({\mathcal {P}}\) is indeed a 2-design.

Remark 7.5

Similarly to the previous example, Proposition 6.2 yields that any 2-design with \(d=5\), \(m_1=1\), and \(m_2=5/3\) must have at least 5 lines, which is matched.

7.6 From 2t-Designs to \(2t+1\)-Designs

Theorem 6.3 yields a construction of \(2t+1\)-designs from 2t-designs. Hence, any 2-design in the Sects. 7.3, 7.4, and 7.5 gives rise to a 3-design. For instance, the 6 lines going through the vertices of the icosahedron are a 2-design in \({\mathbb {R}}^3\). By adding the 6 complementary planes, we obtain a 3-design with respect to \(m_1=m_2=1\).

Remark 7.6

Proposition 6.2 yields that the parts in each single Grassmannian \({\mathcal {G}}_{k,d}\) of a 3-design for \({\mathcal {G}}_{{\mathcal {I}},d}\) with \(|{\mathcal {I}}|=2\) must be a cubature of strength 2. Any such cubature must have at least 6 elements for \(d=3\), so that our 3-design with 12 elements in the union of two Grassmannians is optimal.