Orthogonal and Symplectic Integration

This section details the integration of polynomial functions over the real Orthogonal group $O(d)$ and the Symplectic group $Sp(d)$.

Orthogonal Group $O(d)$

Integration over $O(d)$ differs from the unitary case because the entries $O_{ij}$ are real. Consequently, there is no distinction between $U$ and $\bar{U}$, and we integrate monomials of total even degree.

Theory: Pair Partitions

The integration formula relies on summing over Pair Partitions (also known as matchings) rather than permutations. Let $M_{2n}$ denote the set of all partitions of the set $\{1, \dots, 2n\}$ into $n$ pairs. The size of this set is $(2n-1)!!$.

The general formula is [Collins & Śniady, 2006]:

\[\int_{O(d)} O_{i_1 j_1} \dots O_{i_{2n} j_{2n}} dO = \sum_{\pi_1, \pi_2 \in M_{2n}} \Delta_{\pi_1}(i) \Delta_{\pi_2}(j) \text{Wg}^{O}(\pi_1, \pi_2, d)\]

where:

  • The integrand must have an even total degree ($2n$). If the degree is odd, the integral vanishes.
  • $\Delta_\pi(i)$ is a delta function that equals 1 if the indices $i$ match the pairing $\pi$ (i.e., $i_a = i_b$ for all pairs $\{a, b\} \in \pi$), and 0 otherwise.
  • $\text{Wg}^O(\pi_1, \pi_2, d)$ is the Orthogonal Weingarten function. It depends on the loop structure of the graph formed by superimposing the two matchings $\pi_1$ and $\pi_2$.

1. Basic Integration using @integrate

The @integrate macro automatically identifies O as the random orthogonal matrix.

using IntU, Symbolics
@variables d
# E[O_11^2]
@integrate O[1, 1]^2 dO(d)
# Output: 1/d

2. Manual Integration

For more control, you can declare symbols explicitly.

using IntU, Symbolics
@variables d
O = SymbolicMatrix(:O, :O)
integrate(O[1, 1]^2, dO(d))

3. Higher Moments

The @integrate macro can be used for higher moments too.

using IntU, Symbolics
@variables d
@integrate O[1, 1]^4 dO(d)
# Output: 3 / (d*(d + 2))

Symplectic Group $Sp(d)$

The Symplectic group $Sp(d)$ (Compact Symplectic Group $USp(d)$) consists of unitary matrices that preserve an antisymmetric bilinear form $J$. Crucially, the dimension $d$ must be even ($d=2n$).

The Symplectic Form J

IntU.jl assumes the standard symplectic form $J$:

\[J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}\]

Or in tensor block structure. This matrix is used to relate the conjugate entries to the entries themselves:

\[\bar{S}_{ij} = (J S J^T)_{ij}\]

IntU.jl handles conj(S[i,j]) by automatically substituting it with the appropriate linear combination of $S$ entries and $J$ factors before integration.

Theory

The integration formula is analogous to the orthogonal case but involves the symplectic metric.

\[\int_{Sp(d)} S_{i_1 j_1} \dots S_{i_{2n} j_{2n}} dS = \sum_{\pi_1, \pi_2 \in M_{2n}} \Delta^J_{\pi_1}(i) \Delta^J_{\pi_2}(j) \text{Wg}^{Sp}(\pi_1, \pi_2, d)\]

where $\Delta^J_\pi(i)$ contracts the indices $i$ using the symplectic metric $J$ (introducing signs $\pm 1$). The Weingarten function $\text{Wg}^{Sp}$ is related to $\text{Wg}^O$ by the dimensional shift $d \to -d$ (and sign adjustments).

1. Basic Integration using @integrate

The @integrate macro is also available for the symplectic group. It identifies Sp as the random symplectic matrix.

using IntU, Symbolics
@variables d
# E[|Sp_11|^2]
@integrate abs(Sp[1, 1])^2 dSp(2)
# Output: 1/2

2. Manual Integration

using IntU, Symbolics
@variables d
S = SymbolicMatrix(:S, :Sp)
integrate(abs(S[1, 1])^2, dSp(d))

3. Higher Moments

using IntU, Symbolics
@variables d
@integrate abs(Sp[1, 1])^4 dSp(d)
# Output: 2 / ((d + 1)*(d - 1))

Implementation Details & Pitfalls

  • Automatic Conjugation: For $Sp(d)$, the code treats conj(S) non-trivially. It uses the relation $\bar{S} = -J S J$ to rewrite conjugate entries in terms of $S$ entries (and J factors). This allows using the efficient Weingarten formula for products of $S$ only.
  • Dimension Parity: For $Sp(d)$, $d$ must be even. The symbolic result is valid for even $d$.
  • Computational Complexity: The sum over pair partitions $M_{2n}$ grows as $(2n)!!$. This is significantly faster than $(n!)^2$ for $U(d)$ but still grows combinatorially.
  • Removable Singularities: When using evaluate to substitute numeric values into symbolic results, $0/0$ forms may appear. IntU.jl automatically detects when a denominator evaluates to zero and simplifies the expression to attempt to resolve these removable singularities.

References

  1. Collins, B., & Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic groups. Communications in Mathematical Physics, 264(3), 773-795.
  2. Matsuki, T. (1990). The orbits of affine symmetric spaces under the action of parabolic subgroups. Hiroshima Mathematical Journal. (Relevant for O/Sp symmetry groups).