⇤Math diary
Nov 24, 2023: Polygons, representations, and eigenvalues: The problems
Oof! its been a long time! But I wanna write something down, and my diary lays here open, so what am I to do?
In my pursuits of understanding Hamiltonian G-spaces through the lens of Geometric quantization, I came across a beautiful little story. In this entry, I’m going to present three seemingly unrelated problems with the same answer. In a later entry, I’ll explain why the answers are the same.
Problem 1: Eigenvalues of operators
In linear algebra, there are two main approaches to studying matrices.
- Looking at the entries of the matrix (The additive approach)
- Looking at eigenvalues of the matrix (the multiplicative approach) While the entries of a matrix simply add when adding matrices, they get all mixed up when multiplying or conjugating them. Similarly, eigenvalues multiply when multiplying matrices, and they are preserved under conjugation, but all hell breaks loose when adding them together. This separation is evident using a geometric framework. The additive approach uses lie algebras of matrices, while the multiplicative approach uses the associated lie group (Also matrices). Never the twain shall mix.
Even in the 21st century, we know surprisingly little that crosses this gap. Questions like, “what can you say about the determinant of sums of matrices” are very underdeveloped. One huge open question in matrix analysis is the non-negative inverse eigenvalue problem, “What can we say about the spectrum of a matrix with each entry is nonnegative?”
Here’s something we do know. Consider a hermitian matrix with eigenvalues $\vec\lambda = {\lambda_1,\dots,\lambda_n}$. What are the possible vectors of diagonal entries? We can construct matrices with these eigenvalues by keeping it purely diagonal, with diagonal vector a permutation of ${\lambda_1,\dots,\lambda_n}$. For a permutation $s \in S_n$, We can realize the permuted vector $s \vec{\lambda} \in \mathbb{R}^n$. It turns out these are the most extreme possibilities, everything else is a convex combination:
Theorem: (Schur-Horn): There is a hermitian matrix with eigenvalues ${\lambda_1,\dots,\lambda_n}$ and diagonal entries $\vec d = (d_1,\dots,d_n)$ iff $\vec{d}$ is contained in the convex hull of the permuted eigenvalue vectors, ${s \vec \lambda | s \in S}$.
Here’s a much harder Horn-Klyachko problem: If we know the eigenvalues of a set of hermitian matrices $H_1,\dots,H_{n-1}$, what can we say about the eigenvalues of their sum $H_1+\dots + H_{n-1} = -H_n$? Let us specialize to rank 2 matrices.
Problem: For trace-free hermitian matrices $H_1,\dots,H_{n-1}$ with eigenvalues $\pm \lambda_1, \dots, \pm \lambda_{n-1}$, what are the bounds on the eigenvalue $\pm \lambda_n$ of $H_1+\dots + H_{n-1} = -H_n$?
Problem 2: Clebsch gordon coefficients
For an entirely different field, we are interested in the representations of $SU(2)$. The irreducible representations $V_\lambda \cong \text{Sym}^{\lambda}(\mathbb{C}^2)$ are classified by an integer $\lambda$. These $V_\lambda$ are highest weight representations, with weight $\lambda$. The structure of the representation ring is encompassed by the products coefficients of the irreducible representations. That is, we take the tensor product of two irreps, and decompose them into irreps once again. This gives us an expansion \(V_{\lambda_1} \otimes V_{\lambda_2} = \sum_{\lambda_3 \in \mathbb{Z}} c^{\lambda_3}_{\lambda_1 \lambda_2} V_{\lambda_3}\) The multiplicity $c^{\lambda_3}_{\lambda_1 \lambda_2}$ are called Clebsch-Gordon coefficients. Physicist love these things, because it tells you how two particles with different spin can combine together. For both physics and math, it is important to know exactly which values of $\lambda_1,\lambda_2,\lambda_3$ have $c^{\lambda_3}_{\lambda_1 \lambda_2}$ nonzero. (A physicist would ask something like “can two spin $1/2$ (representation $V_1$) particles combine and create a particle with spin $1/2$ ?” She would then look up the Clebsh-Gordon tables, see that $V_1 \otimes V_1 = V_0 \oplus V_2$, and realize no spin $1/2$ particles are possible.)
We should ask more generally about the decomposition of any number of tensor products
Problem: For which integers ${\lambda_{1}, \dots, \lambda_{n}} \in \mathbb{Z}$ does the tensor product of representations $V_{\lambda_1} \otimes \dots \otimes V_{\lambda_{n-1}}$ have an irreducible component isomorphic to $V_{\lambda_{n}}$?
Problem 3: polygons in 3D
Consider a polygon with $n$ sides embedded in $\mathbb{R}^3$. We can fix the lengths $\lambda_1,\dots,\lambda_n \in \mathbb{R}$ of each side. A polygon is then a collection of vectors ${v_1,\dots,v_n} \in \mathbb{R}^3$ such that $\vert v_i \vert = \lambda_i$ and $\sum v_i = 0$. The moduli space of configurations of these polygons is very geometrically interesting, but let’s not get ahead of ourselves. First we have to ask, when does such a polygon even exist?
Problem: For which ${\lambda_i} \in \mathbb{R}$ does there exist a polygon in $\mathbb{R}^3$ with lengths $\lambda_i$?
Answer
All three problems have the same solution:
Answer: such a $\lambda_1, \dots, \lambda_n$ exist iff for all $j$, \(\lambda_j \leq \sum_{i \neq j} \lambda_i\)
This answer is by far easiest to see for the polygon problem. A polygon with prescribed side lengths exists iff it is not disallowed form the $n$-gon version of the triangle inequality: No side can be longer then the combination of all the others.
This proof identifies the polygon moduli space with the space of operators with given eigenvalues, because they arise in symplectic reduction by stages, with the reduction order swapped. The representation theory question comes from quantizing this moduli space. For a taste of quantization pending future diary entries, see the ‘amusing example’ in the conclusion of this paper