High-dimensional metric-measure limit of Stiefel and flag manifolds

Abstract

We study the high-dimensional limit of (projective) Stiefel and flag manifolds as metric measure spaces in Gromov’s topology. The limits are either the infinite-dimensional Gaussian space or its quotient by some mm-isomorphic group actions, which are drastically different from the manifolds. As a corollary, we obtain some asymptotic estimates of the observable diameter of (projective) Stiefel and flag manifolds.

A. Appendix: \(U^F(N)\) as a subgroup of \(U^\mathbb {R}(N^F)\)

For $z:=z_0+z_1{\mathrm{i}}+z_2{\mathrm{j}}+z_3{\mathrm{k}}\in F^N$, we set

$$\begin{aligned} {{\mathrm{R}}}(z):=z_0,\quad {{\mathrm{I}}}(z):=z_1,\quad {{\mathrm{J}}}(z):=z_2,\quad {{\mathrm{K}}}(z):=z_3. \end{aligned}$$

It follows that for $z:=z_0+z_1{\mathrm{i}}+z_2{\mathrm{j}}+z_3{\mathrm{k}}, w:=w_0+w_1{\mathrm{i}}+w_2{\mathrm{j}}+w_3{\mathrm{k}}\in F^N$,

$$\begin{aligned} \left\langle { \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix}},{\begin{pmatrix} w_0\\ w_1 \\ w_2 \\ w_3 \end{pmatrix}}\right\rangle =\sum _{l=0}^3 z_lw_l ={\mathrm{Re}}\left\langle {z},{w}\right\rangle . \end{aligned}$$

Lemma A.1

Define a map ${\mathcal {O}}^F: U^F(N) \hookrightarrow \mathrm {M}_{N^F}(\mathbb {R})$ by

$$\begin{aligned} {\mathcal {O}}^{F}(U) \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix} := \begin{pmatrix} {{\mathrm{R}}}(Uz)\\ {{\mathrm{I}}}(Uz)\\ {{\mathrm{J}}}(Uz)\\ {{\mathrm{K}}}(Uz)) \end{pmatrix}, \end{aligned}$$

for $z_0,z_1,z_2,z_3 \in \mathbb {R}^N$, where $z:=z_0+z_1{\mathrm{i}}+z_2{\mathrm{j}}+z_3{\mathrm{k}}\in F^N$. Then we have for $U, V \in U^F(N)$,

$$\begin{aligned} {\mathcal {O}}^{F}(U) \in U^\mathbb {R}(N^F), \quad {\mathcal {O}}^{F}(UV)={\mathcal {O}}^{F}(U){\mathcal {O}}^{F}(V), \quad {\mathcal {O}}^{F}(U^*)={\mathcal {O}}^{F}(U)^*. \end{aligned}$$

Proof

For any $z:=z_0+z_1{\mathrm{i}}+z_2{\mathrm{j}}+z_3{\mathrm{k}}, w:=w_0+w_1{\mathrm{i}}+w_2{\mathrm{j}}+w_3{\mathrm{k}}\in F^N$ and $U,V \in U^F(N)$, we compute that

$$\begin{aligned} \left\langle {{\mathcal {O}}^{F}(U) \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix}},{ {\mathcal {O}}^{F}(U) \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix}}\right\rangle ={\mathrm{Re}}\left\langle {Uz},{Uw}\right\rangle ={\mathrm{Re}}\left\langle {z},{w}\right\rangle = \left\langle { \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix}},{\begin{pmatrix} w_0\\ w_1 \\ w_2 \\ w_3 \end{pmatrix}}\right\rangle ,\\ {\mathcal {O}}^{F}(UV) \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix} = \begin{pmatrix} {\mathrm{Re}}(UVz)\\ {{\mathrm{I}}}(UVz)\\ {{\mathrm{J}}}(UVz)\\ {{\mathrm{K}}}(UVz) \end{pmatrix} = {\mathcal {O}}^{F}(U) \begin{pmatrix} {\mathrm{Re}}(Vz)\\ {{\mathrm{I}}}(Vz)\\ {{\mathrm{J}}}(Vz)\\ {{\mathrm{K}}}(Vz) \end{pmatrix} = {\mathcal {O}}^{F}(U) {\mathcal {O}}^{F}(V) \begin{pmatrix} z_0\\ z_1 \\ z_2 \\ z_3 \end{pmatrix}, \end{aligned}$$

implying the first two claims. We also find that ${\mathcal {O}}^{F}(I_N)=I_{N^F}$. This implies that

$$\begin{aligned} {\mathcal {O}}^{F}(U)^*{\mathcal {O}}^{F}(U)=I_N = {\mathcal {O}}^{F}(U^*U) = {\mathcal {O}}^{F}(U^*) {\mathcal {O}}^{F}(U), \end{aligned}$$

hence ${\mathcal {O}}^{F}(U^*)={\mathcal {O}}^{F}(U)^*$. This completes the proof. $\square$