# ATLAS: Tilman Schulz’s integral representations

Tilman Schulz's integral representations of sporadic groups. (Was supposed to include representatives of all complex representations of degree up to 250, but only a few are here.)

For the groups J1 and M11, representatives of all (isomorphism classes of) faithful Q-irreducible representations are given. For J2 only the representation corresponding to the degree 288, 300 and 336 characters are absent (these representations would have degrees 288, 300 and 672, even though all the characters are rational).
Note that the degree 336 character of J2 is a rare example of a character with rational character values and indicator + that cannot be realised over the rationals.

### Contents

#### Group J1

• Dimension 76a over Z: a and b (Magma).
• Dimension 76b over Z: a and b (Magma).
• Dimension 77 over Z: a and b (Magma).
• Dimension 112 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 133 over Z: a and b (Magma).
• Dimension 154 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 209 over Z: a and b (Magma).
• Dimension 266 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 360 over Z: a and b (Magma). — reducible over Q(c19).

#### Group J2

• Dimension 28 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 36 over Z: a and b (Magma).
• Dimension 42 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 63 over Z: a and b (Magma).
• Dimension 90 over Z: a and b (Magma).
• Dimension 126 over Z: a and b (Magma).
• Dimension 140 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 160 over Z: a and b (Magma).
• Dimension 175 over Z: a and b (Magma).
• Dimension 225 over Z: a and b (Magma).
• Dimension 378 over Z: a and b (Magma). — reducible over Q(b5).
• Dimension 448 over Z: a and b (Magma). — reducible over Q(b5).

#### Group J3

• Dimension 170 over Z: a and b (Magma). — reducible over Q(b19).

#### Group M11

• Dimension 10 over Z: a and b (Magma).
• Dimension 11 over Z: a and b (Magma).
• Dimension 20 over Z: a and b (Magma). — reducible over Q(i2).
• Dimension 32 over Z: a and b (Magma). — reducible over Q(b11).
• Dimension 44 over Z: a and b (Magma).
• Dimension 45 over Z: a and b (Magma).
• Dimension 55 over Z: a and b (Magma).

#### Group Th

• Dimension 248 over Z: a and b (Magma).

### Other content

This contains files to check (not prove) the material given here. For each group there is also an output file obtained by checking the representations for the said group.
GroupCheck fileOutput file
J1test.Mtest_out
J2test.Mtest_out
J3test.Mtest_out
M11test.Mtest_out
J1test.Mtest_out

Version 2.0 created on 4th March 2009.
Last updated 04.03.09 by JNB.
R.A.Wilson, R.A.Parker, S.J.Nickerson and J.N.Bray.