Buildings, geometries and algebraic groups study group X

This week Sükrü led the discussion which focussed on root systems, algebraic groups and the Curtis-Tits Theorem.

Recall from an earlier week that given {\alpha\in V=\mathbb{R}^n} and a positive definite bilinear form {(.,.)} on {V}, we can define the reflection

\displaystyle \sigma_{\alpha}:\beta\mapsto \beta-\frac{2(\alpha,\beta)}{(\alpha,\alpha)}\alpha

Then a root system is a subset {\Phi} of {V} such that

  1. {\langle \Phi\rangle=V} and {0\notin \Phi},
  2. If {\alpha\in\Phi} then the only scalar multiples of {\alpha} in {\Phi} are {\pm \alpha},
  3. {\sigma_{\alpha}(\Phi)=\Phi} for all {\alpha\in\Phi},
  4. {2(\alpha,\beta)/(\alpha,\alpha)\in\mathbb{Z}} for all {\alpha,\beta\in\Phi}.

A root system is called irreducible if it cannot be written as a disjoint union of {\Phi_1} and {\Phi_2} such that {\langle \Phi_1\rangle \perp \langle \Phi_2\rangle}.

A fundamental system is then a subset {\Pi} of {\Phi} which is a basis for {V} and such that each element of {\Phi} can be written as a linear combination of the elements of {\Pi} such that the coefficients are either all nonnegative integers or all nonpositive integers.

The Dynkin diagram is the graph with {n} vertices such that the {i^{\mathrm{th}}} vertex is joined to the {j^{\mathrm{th}}} vertex with strength

\displaystyle n_{ij}=\frac{4(\alpha_i,\alpha_j)^2}{\alpha_i,\alpha_i)(\alpha_j,\alpha_j)}\in\mathbb{Z}

Note that {n_{ij}\in\{0,1,2,3\}}. The table below shows what happens for each value of {n_{ij}}.

{n_{ij}} angle between {\alpha_i} and {\alpha_j} name of restriction of diagram to {\{\alpha_i,\alpha_j\}}
{0} {\pi/2} {A_1\times A_1}
{1} {\pi/3} {A_2}
{2} {\pi/4} {B_2}
{3} {\pi/6} {G_2}

Now let {G=\mathrm{SL}_n(\overline{K})}, where {\overline{K}} is an algebraically closed field. A maximal torus {T} is a subgroup maximal subject to being isomorphic to a subgroup of {\overline{K}^{\times}\times\cdots \times \overline{K}^{\times}} and hence can be represented by all diagonal matrices. For each {i,j\in\{1,\ldots,n\}} we can define

\displaystyle  \begin{array}{lcll} \alpha_{ij}:&T&\rightarrow &\overline{K}^{\times}\\ &\left(\begin{array}{ccc} \lambda_1 & &0\\ & \ddots &\\ 0& & \lambda_n\end{array}\right) & \mapsto & \lambda_i\lambda_j^{-1} \end{array}

Let {\Phi=\{\alpha_{ij}\mid i\neq j\}}. The elements {\alpha_{ij}} lie in the dual space {E} of {\overline{K}^n} and we can think of {\Phi} as a subset of {\mathbb{R}\otimes E}, a vector space over {\mathbb{R}}. In this setting {\Phi} is a root system with fundamental system {\Pi=\{\alpha_{12},\alpha_{23},\ldots,\alpha_{n-1,n}\}}.

An alternative way to see these roots is as follows. Let {B} be a Borel subgroup of {G}, that is, a maximal connected solvable subgroup. Then {B=UT} where {U} is unipotent and normal in {B}, while {T} is a torus. We can then choose a minimal {T}-invariant subgroup {U_{\alpha}} of {U}. Then {U_{\alpha}\cong \overline{K}^{+}}. Since {T} normalises {U_{\alpha}} we obtain a map {\alpha:T\rightarrow \mathrm{Aut}(\overline{K}^+)=\overline{K}^\times}.

We can take {B} to be the group of all upper triangular matrices and then {U} is the subgroup of all upper triangular matrices such that the entries on the diagonal are all equal to 1, and {T} is the subgroup of all diagonal matrices. Then there exists {i,j} such that {U_{\alpha}=\{X_{\alpha}(\lambda)=I+\lambda E_{ij}\mid \lambda\in \overline{K}\}}. Now {B} has a unique conjugate {B^-} such that {B\cap B^-=T} and then {B^-=U^-T}. The group {B^-} is the group of all lower triangular matrices.

Note that for {U_{\alpha_1}=\{I+\lambda E_{12}\mid \lambda\in \overline{K}\}} and {U_{-\alpha_1}=\{I+\lambda E_{21}\mid \lambda\in \overline{K}\}} we have {\langle U_{\alpha_1},U_{\alpha_2}\rangle=\mathrm{SL}_2(\overline{K})}.

Now {G=\mathrm{SL}_n(\overline{K})=\langle X_{\alpha}(\lambda)\mid \lambda\in\overline{K},\alpha\in\Phi\rangle}. These generators are known as the Steinberg generators.

We can now state the Curtis-Tits Theorem.

Theorem Let {\Phi} be an irreducible root system of rank at least 3 with fundamental system {\Pi} and Dynkin diagram {\Delta}. Assume that {G} is a group which satisfies the following conditions.

  1. There exist subgroups {K_{\alpha}=\langle U_{\alpha},U_{-\alpha}\rangle} for {\alpha\in \Pi} with {U_{\alpha}\cong U_{-\alpha}\cong K^+} for some field {K} and {K_{\alpha}/Z(K_{\alpha})\cong \mathrm{PSL}_2(K)} such that {G=\langle K_{\alpha}\mid \alpha\in \Pi\rangle}.
  2. {[K_{\alpha},K_{\alpha}]=1} if {\alpha} and {\beta} are not adjacent in the Dynkin diagram.
  3. {\langle K_{\alpha},K_\beta\rangle/Z\cong \mathrm{PSL}_3(K)} if {\alpha} and {\beta} are joined by a single bond.
  4. {\langle K_{\alpha},K_\beta\rangle/Z\cong\mathrm{PSp}_4(K)} if {\alpha} and {\beta} are joined by a double bond.
  5. {N_{K_{\alpha}}(U_{\alpha})\cap N_{K_{\alpha}}(U_{-\alpha})\leqslant N_G(U_{\beta})} for all {\alpha,\beta\in \Pi}.

Then there exists a group of Lie type {\tilde{G}} with corresponding root system {\Phi}, fundamental system {\Pi} and Dynkin diagram {\Delta} and a surjective homomorphism from {G} onto {\tilde{G}}.

We note it is possible to have {K_{\alpha}} isomorphic to {\mathrm{SL}_2(K)} or {\mathrm{PSL}_2(K)}. For example, for groups of Lie type with root systems of type {B_n}, the long roots all have {K_{\alpha}\cong \mathrm{SL}_2(K)} while the short root has {K_{\alpha}\cong \mathrm{PSL}_2(K)}.

Sükrü will continue next week. He is also giving a Groups and Combinatorics seminar on Tuesday on related material.

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