THEOREM. Let X_{H,V} be a stable summand of (BH_{+})_{2}^{∧} corresponding to the simple 𝔽_{2} Out(H)-module V, where H is a 2-group. The matrix

rank_{ℤ_{p}} Hom(X_{H,V} , X_{K,W})

is the Cartan matrix shown, where X = all and Y = 1.

THEOREM. Over any field R, for any class of groups X closed under extensions and sections, the Cartan matrix C_{R}^{X ,X} of globally defined Mackey functors Mack_{R}^{X ,X} is symmetric and non-singular.

THEOREM. Let (K, O, k) be a p-modular system. Then the Cartan matrices and decomposition matrices satisfy C_{k} = D^{T} C_{K}D.

The table of values S_{H,V} (G) of simple functors in Mack_{ℂ}^{1,1}. Because the characteristic is zero we have S_{H,V} = Δ_{H,V}. These are also the values of the simple functors in Mack_{ℂ}^{all,1}.

Formal characters S_{H,V}
X = all or 1, Y = 1 1 C_{2} C_{4}  C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 0  1 1 1 −1 1 −1 2 1 −1_{3} −1_{5} −1_{7} 1 −1 2 1 −1
K, W  1 1 1
C_{2} 1 1 1
C_{4} 1 1 1 1
−1  1
C_{2} × C_{2} 1 1 1  1
−1  1
2 1 1
C_{8} 1 1 1 1  1
−1_{3}  1
−1_{5} 1 1
−1_{7}  1
Q_{8} 1 1 1 1  1
−1  1
2  1  1
D_{8} 1 1 2 1  1  1  1
−1  1  1  1  1

Table of values of simple functors in Mack_{𝔽_{2}}^{1,1}.

Formal characters  S_{H,V}
X = 1, Y = 1  1  C_{2}  C_{4} C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 2  1  1  1  1 2  1  1 2  1
K, W  1 1  1
C_{2} 1  1
C_{4} 1  1
C_{2} × C_{2} 1  1
2  1
C_{8} 1  1
Q_{8} 1  1
2  1
D_{8} 1  2  1

The Cartan matrix for Mack_{ℂ}^{1,1} in characteristic 0 is the identity matrix Decomposition matrix for Mack_{ℂ}^{1,1} from characteristic 0 to characteristic 2:

Decomposition matrix S_{H,V}
X = 1, Y = 1  1  C_{2}  C_{4} C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 2  1  1  1  1 2  1  1 2  1
S_{K,W}  1  1  1  1  1  1  1  1  1
C_{2}  1  1  1  1 1  1  1  1
C_{4}  1  1  1  1 1  1
−1  1  1
C_{2} × C_{2}  1  1  2
−1  1
2  1
C_{8} 1 1
−1_{3}  1
−1_{5}  1
−1_{7}  1
Q_{8} 1 1
−1  1
2  1
D_{8} 1 1
−1  1

Cartan matrix C_{𝔽_{2}} = D^{T} D for Mack_{𝔽_{2}}^{1,1} in characteristic 2:

Cartan matrix  P_{H,V}
X = 1, Y = 1  1  C_{2}  C_{4} C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 2  1  1  1  1 2  1  1 2  1
S_{K,W}  1 1  1  1  1  1 0  1  1 0  1
C_{2} 1  1  2  2  2 1  2  2 0  2
C_{4} 1  1  2  4  2 1  4  3 1  3
C_{2} × C_{2} 1  1  2  2  4 1  2  2 0  4
2  0  1  1  1 2  1  1 0  1
C_{8} 1  1  2  4  2 1  8  3 1  3
Q_{8} 1  1  2  3  2 1  3  5 1  3
2  0  0  1  0 0  1  1 2  1
D_{8} 1  1  2  3  4 1  3  3 1  9

Cartan matrix C_{ℂ} for Mack_{ℂ}^{all,1} in characteristic 0:

Cartan matrix  P_{H,V}
X = all, Y = 1  1 C_{2} C_{4}  C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 0  1 1 1 −1 1 −1 2 1 −1_{3} −1_{5} −1_{7} 1 −1 2 1 −1
S_{K,W}  1 1 1
C_{2} 1 1 1
C_{4} 1 1 1 1
−1  1
C_{2} × C_{2} 1 1 1  1
−1  1
2 1 1
C_{8} 1 1 1 1  1
−1_{3}  1
−1_{5}  1
−1_{7}  1
Q_{8} 1 1 1  1  1
−1  1  1
2  1  1  1
D_{8} 1 1 2  1  1  1
−1  1  1  1

Cartan matrix C_{𝔽_{2}} = D^{T} C_{ℂ}D for Mack_{𝔽_{2}}^{all,1} in characteristic 2:

Cartan matrix  P_{H,V}
X = all, Y = 1  1  C_{2}  C_{4} C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 2  1  1  1  1 2  1  1 2  1
S_{K,W}  1 1  1  0  0  0 0  0  0 0  0
C_{2} 1  1  1  1  1 0  1  1 0  1
C_{4} 1  2  2  4  2 0  4  3 1  3
C_{2} × C_{2} 1  2  2  2  4 0  2  2 0  4
2  0  1  1  1 1  1  1 0  1
C_{8} 1  3  3  6  3 0  10  5 2  5
Q_{8} 1  3  3  4  5 0  4  6 1  6
2  1  2  3  2 1  3  3 2  3
D_{8} 1  5  7  8  10 2  8  8 1  16

Decomposition matrix for Mack_{ℂ}^{all,1} from characteristic 0 to characteristic 2:

Decomposition matrix S_{H,V}
X = all, Y = 1  1  C_{2}  C_{4} C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 2  1  1  1  1 2  1  1 2  1
S_{K,W}  1  1  1
C_{2}  1  1  1  1  1  1  1
C_{4}  1  1  1  1 1  1
−1  1  1
C_{2} × C_{2}  1  1  2
−1  1
2  1
C_{8} 1 1
−1_{3}  1
−1_{5}  1
−1_{7}  1
Q_{8} 1 1
−1  1
2  1
D_{8} 1 1
−1  1

Table of values of simple functors in Mack_{𝔽_{2}}^{all,1}:

Formal characters  S_{H,V}
X = all, Y = 1  1  C_{2}  C_{4} C_{2} × C_{2} C_{8}  Q_{8}  D_{8}
characteristic 2  1  1  1  1 2  1  1 2  1
K, W  1 1  1
C_{2} 1  1  1
C_{4} 1  1  1
C_{2} × C_{2} 1  1  1
2  1  1
C_{8} 1  1  1
Q_{8} 1  1  1
2  1
D_{8} 1  1  2  2  1