Modular \SL_2 characters

Let G = \SL_2 be the special linear group, defined over a field of characteristic p > 0. The character ring of \SL_2 consists of symmetric Laurent polynomials p(v): symmetric meaning that p(v) = p(v^{-1}). There are several natural bases of this ring, each indexed by the natural numbers, which will be explained further below:

  1. The monomial (squares) symmetric functions m_0 = 1 and m_n = v^n + v^{-n}.
  2. The Weyl (triangles) characters \chi(n) = \ch \Delta(n) = v^n + v^{n-2} + \cdots + v^{2-n} + v^{-n}.
  3. The simple (circles) characters \ch L(n), where L(n) is the simple module with highest weight n.
  4. The tilting (stars) characters \ch T(n), where T(n) is the indecomposable tilting module with highest weight n.

The monomial functions and Weyl characters do not depend on the characteristic p, but the simple and tilting characters do. The following visualisation shows the transition matrices between each of these bases. Blue means positive multiplicity, and red means negative. A larger absolute value of multiplicity will give a larger shape.

Monomial symmetric functions

Let X be the character lattice, then the character of a representation is valued in \mathbb{Z}[X]^W, the W-invariant subring of the group algebra \mathbb{Z}[W]. In the case of \SL_2, we have X = \mathbb{Z}, where we write the basis element of \mathbb{Z}[X] corresponding to n \in X as v^n. The Weyl group W = \innprod{s} \subseteq \GL_\bbZ(X), where s(n) = -n is the reflection over the origin. It follows that a character (i.e. a Laurent polynomial) p \in \bbZ[X] is invariant under W if and only if p(v) = p(v^{-1}).

When a group is acting on a set X, then the W-invariant subspace inside the free \bbZ-module with basis X has an obvious basis: the sum over the orbits. Hence we define the monomial symmetric functions m_\lambda for any \lambda \in X to be m_\lambda = \sum_{\mu \in W \lambda} v^\mu. The set of all m_\lambda is not linearly independent, for example in our case we have m_{-3} = m_3 = v^3 + v^{-3}. We need to choose a representative from each orbit to narrow the set down to a basis. The dominant weights X^+ = \set{\lambda \in X \mid \innprod{\alpha^\vee, \lambda} \geq 0 \text{ for all } \alpha \in \Phi^+} \subseteq X will work. In our case, the dominant weights are exactly the natural numbers. Hence we get our monomial basis \{m_0, m_1, m_2, \ldots\}.

Weyl characters

The Weyl character \chi(n) = \ch \Delta(n) is the character of a Weyl module, which coincides with the simple highest-weight module of a semisimple Lie algebra. For \SL_2 the character \chi(n) is quite simple, starting at v^n and hopping down by 2 = \alpha each time until reaching v^{-n}:

\chi(n) = v^{n} + v^{n - 2} + \cdots + v^{2-n} + v^{-n}.

The reader may also know this character either by the Weyl character formula, or as a symmetric quantum integer:

\chi(n) = \frac{\sum_{w \in W} (-1)^w v^{w(n + \rho)}}{\sum_{w \in W} (-1)^w v^{w \rho}} = \frac{v^{n + 1} - v^{-n -1}}{v - v^{-1}} = [n+1]_v.

One fact we will need about the Weyl characters is that the expression \chi(n) makes sense for n < 0, and furthermore satisfies \chi(-1) = 0 and \chi(n) = -\chi(s \bullet n) for the simple reflection s, where \bullet denotes the shifted group action w \bullet \lambda = w(\lambda + \rho) - \rho, where \rho = 1 is the sum of the fundamental weights.

Simple characters

The simple module L(n) depends on the characteristic p, and can be defined as the head of the Weyl module \Delta(n), or as the socle of the induced module \nabla(n). It is usually difficult to write down the characters of the simple modules in characteristic p > 0, but for \SL_2 we can do it. The simple characters \ch L(n) are determined by the residues of binomial coefficients modulo p: we have that v^{n - i \alpha} appears in \ch L(n) if and only if \binom{n}{i} is nonzero modulo p. I have a visualisation of these residues.

One way to see this is to calculate the Shapovalov form on the highest-weight module \Delta(n): on the n - i\alpha weight space, the form evaluates as \binom{n}{i}. The radical of this form gives the maximal submodule of \Delta(n) (this is not obvious!), and the result follows.

Tilting characters

Even less obvious than the characters of simples in characteristic p are the characters of the tilting modules. We have that T(n) = \Delta(n) for 0 \leq n < p (i.e for n \in \overline{C}_\mathbb{Z}, this is almost “for free”), and then an \SL_2-specific calculation shows that \ch T(n) = \chi(n) + \chi(t^{(1)} \bullet n) for p \leq n < 2p - 1, where t^{(1)} is the p-dialated affine simple reflection, i.e. t^{(1)} is the affine reflection over the hyperplane through p, and \bullet is the \rho-shifted action as above.

After this, the rest of the tilting character table can be filled out by using the Donkin tilting tensor product theorem, which states that for \lambda \in X_1 + (p - 1)\rho and any other dominant weight \mu we have T(\lambda + p \mu) = T(\lambda) \otimes T(\mu)^{(1)}. (This theorem is somewhat remarkable because the Frobenius twist of a tilting module is not tilting!).

For example, suppose we want to determine T(10) for p = 3. First we subtract p - 1 to get 8, then write 8 in its base-3 form 22_3, then add p-1 back on to get 10 = 24_3. This means that we have 10 = 4 + 2p, and p-1 \leq 4 \leq 2p - 2, so we can apply the Donkin tensor product formula: we need to know \ch T(4) = \chi(4) + \chi(0) and \ch T(2)^{(1)} = \chi(2)^{(1)} = (v^2 + 1 + v^{-2})^{(1)} = v^6 + 1 + v^{-6}. One can multiply \chi’s and v’s by \chi(a) v^b = \chi(a + b) so long as we’re multiplying by a symmetric sum of v’s: we have then

\begin{aligned} \ch T(10) &= (\chi(4) + \chi(0))(v^6 + 1 + v^{-6}) \\ &= \chi(10) + \chi(-2) + \chi(4) + \chi(0) + \chi(6) + \chi(-6) \\ &= \chi(10) - \chi(0) + \chi(4) + \chi(0) + \chi(6) - \chi(4) \\ &= \chi(10) + \chi(6) \\ \end{aligned}

In the case of \SL_2, iterating this rule allows us to get any tilting character just by knowing the first 2p - 1 tilting characters.