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The decomposition matrix of S3 in characteristic two

Example

For $S_{3}$ in characteristic $2$ , with ordinary irreducibles ordered as $(1, sgn, std)$ and irreducible Brauer characters ordered as $(φ_{t r i v}, φ_{s t d})$ , the decomposition matrix is

$(\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}) .$

Facts & Assumptions

Given: The ordinary trivial, sign, and standard characters of $S_{3}$ in characteristic $2$ .

[F1]

The decomposition matrix is defined by the expansions of the restricted ordinary characters in the irreducible Brauer basis (Decomposition numbers and the decomposition matrix).

[F2]

The decomposition map records those modular reductions (Decomposition map from ordinary to modular Grothendieck groups).

[L1]

The $2$ -regular classes of $S_{3}$ are represented by $1$ and $(123)$ (The p-regular classes of S3).

Verification

technique · direct

1.1

By [L1], only the values at $1$ and $(123)$ matter. The ordinary trivial and sign characters both restrict to $(1, 1)$ on those classes, because a $3$ -cycle is even. The ordinary standard character restricts to $(2, - 1)$ .

L1givenalgebra

2.1

In characteristic $2$ , the trivial module gives the Brauer character $φ_{t r i v} = (1, 1)$ , while the $2$ -dimensional simple module gives $φ_{s t d} = (2, - 1)$ . Therefore [F1] writes the restricted ordinary characters as $1^{0} = φ_{t r i v}, {sgn}^{0} = φ_{t r i v}, {std}^{0} = φ_{s t d} .$

F1F2step 1.1algebra

3.1

Reading off the coefficients yields the displayed decomposition matrix.

step 2.1∎

Depends on

Used by

A Cartan matrix computed from D^T D Example

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)