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B and C are always isomorphic

Statement

False for $n \geq 3$ : $B_{n}$ and $C_{n}$ are dual to one another but are not isomorphic root systems; only $n \leq 2$ gives an isomorphism.

Facts & Assumptions

Given: The standard coordinate models $B_{n} = {\pm ε_{i}, \pm ε_{i} \pm ε_{j} : i < j}$ and $C_{n} = {\pm 2 ε_{i}, \pm ε_{i} \pm ε_{j} : i < j}$ in $R^{n}$ .

[L1]

These are the root systems of types $B_{n}$ and $C_{n}$ , with the squared lengths $1$ and $2$ in $B_{n}$ and $2$ and $4$ in $C_{n}$ (Existence of each classified root system, Root systems of the classical complex Lie algebras).

[L2]

An isomorphism of root systems preserves all Cartan integers, hence preserves angles and the ratios of lengths; in an irreducible system it therefore maps roots of maximal length to roots of maximal length and roots of minimal length to roots of minimal length (Rank and isomorphism of root systems, Rank-two root-system classification).

[L3]

$B_{2} ≅ C_{2}$ , while for $n \geq 3$ the types are distinct in the classification list (Duality exchanges B and C, Classification of irreducible root systems).

Refutation

technique · counterexample

1.1

In $B_{n}$ the roots of squared length $2$ are the $\pm ε_{i} \pm ε_{j}$ with $i < j$ , of which there are $2 n (n - 1)$ , and the roots of squared length $1$ are the $\pm ε_{i}$ , of which there are $2 n$ . In $C_{n}$ the roles are exchanged: the roots of squared length $4$ are the $\pm 2 ε_{i}$ , of which there are $2 n$ , and the roots of squared length $2$ are the $\pm ε_{i} \pm ε_{j}$ , of which there are $2 n (n - 1)$ .

L1algebra

2.1

An isomorphism $B_{n} \to C_{n}$ would preserve the length classes by [L2], so it would carry the $2 n (n - 1)$ long roots of $B_{n}$ bijectively onto the $2 n$ long roots of $C_{n}$ and the $2 n$ short roots of $B_{n}$ onto the $2 n (n - 1)$ short roots of $C_{n}$ ; for $n \geq 3$ these cardinalities differ, since $2 n (n - 1) > 2 n$ . Hence $B_{n} ≇ C_{n}$ for $n \geq 3$ . For $n = 1$ , the coordinate models in [L1] are $B_{1} = {\pm ε_{1}}$ and $C_{1} = {\pm 2 ε_{1}}$ , and the linear map $x \mapsto 2 x$ carries one onto the other and preserves their sole Cartan integer $2$ . For $n = 2$ the systems are isomorphic by [L3]. Therefore the isomorphism holds exactly for $n \leq 2$ .

L1L2L3step 1.1algebra∎

Depends on

Used by

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)