Conventions

In these notes and during classtime we will adhere to the following conventions.

We use the standard notation \(\mathbb{Z}\) and \(\mathbb{Z}^+\) to stand for the sets of integers and positive integers, respectively. Perhaps controversially, we will use \(\mathbb{N}\) to stand for the non-negative integers (that is, in these notes we say that \(0 \in \mathbb{N}\)). We use \(\mathbb{Q}\) and \(\mathbb{Q}^+\) for the sets of rationals and positive rationals, respectively, and similarly write \(\mathbb{R}\) and \(\mathbb{R}^+\) for the sets of reals and positive reals, respectively. We also denote the complex numbers by \(\mathbb{C}\).

Capitalized letters will be used to stand for sets, possibly with additional structure, and for linear maps. The letters \(G, H, K\) will be used for groups, where the latter might specifically indicate the kernel of a homomorphism when made clear from context. The letter \(N\) will often be used to denote a normal subgroup, especially if it is not evidentally the kernel of some homomorphism. The letters \(R\) and \(S\) will stand for rings and algebras; \(F\) and \(E\) for fields; \(U\), \(V\), and \(W\) for vector spaces; \(X\) and \(Y\) will stand for sets, perhaps beset upon by a group action. The letters \(B\) and \(C\) stand in reserve for algebraic objects if we exhaust other options, (e.g., where the use of \(K\) might be confused with a kernel). In addition, letters like \(A, B, L, M, P, Q,\) and \(T\) refer to operators.

Lowercase letters like \(n, m,\) and \(\ell\) will be reserved for integers; \(i, j\), and \(k\) will be used for indices. Note that \(i, j,\) and \(k\) might be used for imaginary numbers or quaternions, though again such context will always be clear. Letters like \(g, h,\) and \(k\) will be used to stand for group elements. If more than one group is present, such as \(G\) and \(H\), lower case letters will be used harmoniously with upper case letters: \(g \in G\), \(h \in H\), and so on. Similarly, \(r\) and \(s\) will be used for ring elements. We may use primes or indices to denote multiple elements in the same group, e.g., \(g, g' \in G\) or \(g_1, g_2 \in G\), when other groups are present to make something like \(g, h \in G\) unnecessarily confusing.

Note that \(x\) and \(y\) can sometimes be used to stand for elements from arbitrary sets, perhaps those under the influence of a group action, or as indeterminants in a polynomial ring. In linear algebra, we use \(x, y,\) and \(z\) to stand for vectors and \(u, v\), or \(w\), decorated with subscripts, to stand for basis elements. The usual \(e_i\) notation is used for the standard basis elements of \(F^n\). We will use \(t\) to denote a real parameter, with \(s\) as a potential auxiliary, and also as the indeterminant for characteristic polynomials.

The Greek letters \(\rho, \sigma,\) and \(\tau\) will be used for group homomorphisms; \(\varepsilon\) stands for the sign representation \(\mathcal{S}_n \to \mathcal{Z}_2 = \{1, -1\}\). We will also use \(\sigma\) and \(\tau\) to denote permutations if there is no danger of confusion. Note that \(\zeta\) is sometimes used to stand for a root of unity, especially \(\zeta_n \coloneqq e^{2\pi i/n}\) as a primitive \(n\)th root. As is customary, \(\chi\) is used for characters of representations. The letters \(\varphi\) and \(\psi\) are used especially for intertwiners; \(\mu\) is used to indicate a bi-linear form. Meanwhile, \(\alpha, \beta, \gamma,\) and \(\lambda\) are scalars, where the latter is used especially to indicate an eigenvalue.

We sometimes use the hooked arrow \(\hookrightarrow\) to denote a one-to-one function, especially an inclusion: e.g., \(X \hookrightarrow Y\). Dually, the double-headed \(\twoheadrightarrow\) is used to emphasize that a map is onto. We write \(\mathrm{id}_X\) for the identity map \(X \to X\), or simply \(\mathrm{id}_X\) when context is clear. Lastly, we write \[ X \xrightarrow{\cong}Y \] for bijections (or isomorphisms, if a category is clear) or simply \(X \leftrightarrow Y\) when directionality is unimportant.