2024-01-17--notes-RT.md

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title: (Representation Theory) Notes on Groups & Linear Algebra
date: 2024-01-17

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In the first lecture, we discussed some examples of groups and some basics of linear algebra.

Groups

• the elements of the cylic group $\mathbb{Z}/n\mathbb{Z}$ are the equivalence classes of integers under the relation "$\equiv \pmod{n}$"

  this group is additive

• we observed that the mapping $\phi:\mathbb{R} \to \mathbf{S}^1$ given by $\phi(t) = e^{2\pi i t}$ is a group homomorphism since $\phi(t + s) = \phi(t) \phi(s)$ for all $t,s \in \mathbb{R}$.

  we observed that $\ker \phi = \mathbb{Z}$, and that - by the First Isomorphism Theorem - $\phi$ induces an isomorphism $$\overline{\phi}: \mathbb{R}/\mathbb{Z} \to \mathbf{S}^1.$$

• for a non-zero natural number the symmetric group $S_n$ is the collection of all bijections $I_n \to I_n$ where $I_n = \{1,2,\cdots,n\}$.

  We may sometimes use cycle notation for elements of $S_n$.

  The subgroup $$H=\langle (1234), (14)(23) \rangle$$ has order $8$ and is sometimes called the dihedral group $D_4$ or $D_8$ -- it has order 8.

• Let $F$ be a field.

  Recall that typically examples are: $F = \mathbb{R},\mathbb{C},\mathbb{Q},\mathbb{Z}/p\mathbb{Z}$ for a prime number $p$.

  The set $$\operatorname{GL}_n(F) = \{\text{all invertible $n \times n$matrices with entries in$FParseError: KaTeX parse error: Expected 'EOF', got '}' at position 1: }̲\}$ forms a group under matrix multiplication.

  The determinant function yields a group homomorphism $$\det:\operatorname{GL}_n(F) \to F^\times$$ (here $F^\times$ means $F \setminus \{0\}$, which is a commutative group under multiplication in the field $F$).

Linear Algebra

Let $F$ be a field. An $F$-vector space $V$ is an additive abelian group together with an operation of scalar multiplication -- this amounts to a function $$F \times V \to V$$ -- satisfying certain axioms.

If $V,W$ are $F$-vector spaces, a linear mapping $T:V \to W$ is a function which satisfies $$T(\alpha v + w) = \alpha T(v) + T(w).$$

Let's suppose that $V$ is finite dimensional and that $\phi:V \to V$.

We write $\phi^2 = \phi \circ \phi$ and more generally $\phi^n = \phi \circ \phi^{n-1}$.

trace, det, char poly

The trace of a matrix $M=[M_{ij}]$ is the sum of the diagonal entries: $$\operatorname{tr}(M) = \sum_{i=1}^n M_{ii}.$$

I'm assuming you recall the definition of the determinant $\det M$.

The characteristic polynomial $\operatorname{cp}_M(X) \in F[X]$ of $M$ is defined to be $$\operatorname{cp}_M(X) = \det(M - X \cdot \mathbf{I}_n).$$

For a linear transformation $\phi$ we define

• $\operatorname{tr}(\phi) = \operatorname{tr}([\phi]_{\mathcal{B}})$

• $\operatorname{det}(\phi) = \operatorname{det}([\phi]_{\mathcal{B}})$

• $\operatorname{cp}_\phi(X) = \operatorname{cp}_{[\phi]_{\mathcal{B}}}(X)$

Proposition : $\operatorname{tr}(\phi)$, $\operatorname{det}(\phi)$, and $\operatorname{p}_\phi(X)$ are independent of the choice $\mathcal{B}$ of basis for $V$.

The main point here is that if $\mathcal{B}$ and $\mathcal{B}'$ are two basis for $V$, there is an invertible matrix ("change of basis matrix") $P$ for which $$[\phi]_{\mathcal{B}} = P [\phi]_{\mathcal{B}'} P^{-1}.$$

Evaluation of polynomials at linear transformations

Suppose that $f = f(X) \in F[X]$ is a polynomial; thus $$f = \sum_{i=0}^N a_i X^i$$ for some coefficients $a_i \in F$.

We may evaluate the polynomial $f$ at the linear endmorphism $\phi$:

$$f(\phi) = \sum_{i=0}^N a_i \phi^i.$$

Proposition : Let $\phi:V \to V$ be a linear transformation, and let $$I = \{f \in F[X] \mid f(\phi) = 0\}.$$ Then $I$ is an ideal in the polynomial ring $F[X]$. In particular, there is a unique monic polynomial $m_\phi(X) \in F[x]$ for which $I = m_\phi(X) F[X]$.

In particular, if $f \in F[X]$ and $f(\phi)=0$, then $m_\phi \vert
f$.

Theorem (Cayley-Hamilton) : Let $\phi:V \to V$ be a linear transformation, and let $\operatorname{cp}(X) = \operatorname{cp}_\phi(X) \in F[X]$ be the characteristic polynomial.

Then $\operatorname{cp}(\phi) = 0$.

Recall that the eigenvalues of $\phi$ are precisely the roots of the characteristic polynomial. The Cayley-Hamilton Theorem implies that any root of the minimal polynomial is an eigenvalue. In fact, we have the converse as well:

Proposition: : If $\lambda \in F$ is an eigenvalue of $\phi$ -- i.e. a root of the characteristic polynomial -- then $\lambda$ is a root of the minimal polynomial.

Theorem: : $\phi$ is diagonalizable -- i.e. $V$ has a basis of eigenvectors for $\phi$ -- if and only the minimal polynomial has no multiple roots.

Remark: : This theorem should be proved in Math215-216 here at Tufts using the Fundamental Theorem for modules over a PID. We don't need the full force of this result in our class.

Example

Suppose that $\phi:V \to V$ satisfies $\phi^N = \operatorname{id}_V$ for some positive natural number $N$.

We suppose that $F$ is algebraically closed and of characteristic zero.

Notice that the polynomial $f(X) = X^N -1 \in F[X]$ has distinct roots.

(If $F = \mathbb{C}$, these roots are exactly $$\{\exp(2\pi k i/n) \mid 0 \le k < N\}.$$ )

Since the minimal polynomial of $\phi$ divides $f$, we see that the minimal polynomial has distinct roots and hence $\phi$ is diagonalizable by the Theorem quoted above.