feat(RepresentationTheory): add stabilizers of vectors in representations

[JX-Mo]

Add Representation.stabilizer: the subgroup fixing a vector in a representation.

Provide basic lemmas for zero vectors, scalar multiples, sums, intertwining maps, and conjugations, as a first step toward smooth representations. Make representation-theoretic arguments use this specialized API instead of repeatedly unfolding the fixed-vector condition or routing through the more general MulAction stabilizer API.

[github-actions]

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[github-actions]

PR summary 6215b7002a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RepresentationTheory.Stabilizer (new file)	1482

---

Declarations diff (regex)

+ IntertwiningMap.stabilizer_le
+ le_stabilizer_add
+ le_stabilizer_smul
+ le_stabilizer_sum
+ mem_stabilizer
+ stabilizer
+ stabilizer_conj
+ stabilizer_zero

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 6215b70).

• +8 new declarations
• −0 removed declarations

+Representation.IntertwiningMap.stabilizer_le
+Representation.le_stabilizer_add
+Representation.le_stabilizer_smul
+Representation.le_stabilizer_sum
+Representation.mem_stabilizer
+Representation.stabilizer
+Representation.stabilizer_conj
+Representation.stabilizer_zero

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
4974	1	exposed public sections

Current commit 6215b7002a
Reference commit 7bc792d598

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[tb65536]

Remember to remove the "awaiting-author" tag once you are done responding to all of the comments (type "-awaiting-author")

[JX-Mo]

-awaiting-author

[JX-Mo]

"-awaiting-author"

[JX-Mo]

My subsequent PR #41199 about basics of smooth representations is now available on GitHub.