Vipul at 19:35, 21 May 2012

2012-05-21T19:35:23Z

← Older revision		Revision as of 19:35, 21 May 2012
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	group = Conway group:Co0\|		group = Conway group:Co0\|
	connective = of}}		connective = of}}

			==Summary==

			<section begin="summary"/>
			{\| class="sortable" border="1"
			! Item !! Value
			\|-
			\| [[degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) \|\| too long to list, see [[Linear representation theory of Conway group:Co0#GAP implementation]]<br>[[number of irreducible representations equals number of conjugacy classes\|number]]: 167, [[sum of squares of degrees of irreducible representations equals order of group\|sum of squares]]: 8315553613086720000, [[maximum degree of irreducible representation\|maximum]]: 1021620600, [[quasirandom degree]]: 24
			\|}
			<section end="summary"/>

			==Irreducible representations==

			===Smallest degree nontrivial irreducible representation===

			We know that the Conway group is the automorphism group of the [[Leech lattice]]. All automorphisms of a lattice are ''linear'' automorphisms, hence this action defines a linear representation of the Conway group <math>\operatorname{Co}_0</math> on a 24-dimensional representation over the real numbers. This representation is irreducible over the complex numbers and is the smallest degree nontrivial irreducible representation.

			Note that the representation is faithful, and it does ''not'' descend to an irreducible representation of [[Conway group:Co1]].

			Some information on its irreducible representations is, and we use the symbol "2.Co1" to access this information. The degrees of irreducible representations can be computed using the [[GAP:CharacterDegrees\|CharacterDegrees]] and [[GAP:CharacterTable\|CharacterTable]] functions.

			<pre>gap> CharacterDegrees(CharacterTable("2.Co1"));
			[ [ 1, 1 ], [ 24, 1 ], [ 276, 1 ], [ 299, 1 ], [ 1771, 1 ], [ 2024, 1 ],
			[ 2576, 1 ], [ 4576, 1 ], [ 8855, 1 ], [ 17250, 1 ], [ 27300, 1 ],
			[ 37674, 1 ], [ 40480, 1 ], [ 44275, 1 ], [ 80730, 1 ], [ 94875, 1 ],
			[ 95680, 1 ], [ 170016, 1 ], [ 299000, 1 ], [ 313950, 1 ], [ 315744, 1 ],
			[ 345345, 1 ], [ 351624, 1 ], [ 376740, 1 ], [ 388080, 1 ], [ 483000, 1 ],
			[ 644644, 1 ], [ 673750, 2 ], [ 789360, 1 ], [ 822250, 1 ], [ 871884, 1 ],
			[ 1434510, 1 ], [ 1450449, 1 ], [ 1771000, 1 ], [ 1821600, 1 ],
			[ 1841840, 1 ], [ 1937520, 1 ], [ 2055625, 1 ], [ 2417415, 1 ],
			[ 2464749, 2 ], [ 2816856, 1 ], [ 2877875, 1 ], [ 4004000, 1 ],
			[ 4100096, 1 ], [ 5051904, 1 ], [ 5494125, 1 ], [ 5801796, 1 ],
			[ 6446440, 2 ], [ 7104240, 1 ], [ 7628985, 1 ], [ 9152000, 2 ],
			[ 9221850, 1 ], [ 9669660, 1 ], [ 11051040, 1 ], [ 12432420, 1 ],
			[ 13156000, 1 ], [ 15002624, 2 ], [ 15471456, 1 ], [ 16170000, 2 ],
			[ 16347825, 1 ], [ 17050176, 1 ], [ 17310720, 1 ], [ 18987696, 1 ],
			[ 19734000, 1 ], [ 20083140, 1 ], [ 21049875, 1 ], [ 21528000, 1 ],
			[ 21579129, 1 ], [ 23244375, 1 ], [ 24174150, 1 ], [ 24667500, 1 ],
			[ 24794000, 1 ], [ 25900875, 1 ], [ 31574400, 1 ], [ 34155000, 1 ],
			[ 40166280, 1 ], [ 40370176, 3 ], [ 44013375, 1 ], [ 44204160, 1 ],
			[ 46621575, 1 ], [ 49335000, 1 ], [ 50519040, 1 ], [ 51571520, 2 ],
			[ 55255200, 1 ], [ 57544344, 1 ], [ 59153976, 3 ], [ 60435375, 1 ],
			[ 62790000, 1 ], [ 65270205, 1 ], [ 66602250, 1 ], [ 67358720, 1 ],
			[ 77702625, 1 ], [ 83720000, 1 ], [ 85250880, 1 ], [ 91547820, 1 ],
			[ 100725625, 1 ], [ 106142400, 1 ], [ 106260000, 1 ], [ 109882500, 1 ],
			[ 112519680, 1 ], [ 139243104, 1 ], [ 150732800, 1 ], [ 161161000, 1 ],
			[ 163478250, 1 ], [ 184184000, 1 ], [ 185912496, 1 ], [ 185955000, 1 ],
			[ 190417920, 1 ], [ 191102976, 1 ], [ 201451250, 1 ], [ 205395750, 1 ],
			[ 207491625, 1 ], [ 210496000, 1 ], [ 210974400, 1 ], [ 215547904, 2 ],
			[ 219648000, 1 ], [ 230230000, 1 ], [ 241741500, 1 ], [ 247235625, 1 ],
			[ 247543296, 1 ], [ 251756505, 1 ], [ 257857600, 1 ], [ 259008750, 1 ],
			[ 267014475, 1 ], [ 280280000, 1 ], [ 282906624, 1 ], [ 287006720, 1 ],
			[ 292953024, 1 ], [ 299710125, 1 ], [ 302176875, 1 ], [ 309429120, 1 ],
			[ 313524224, 1 ], [ 326956500, 1 ], [ 342752256, 1 ], [ 351624000, 1 ],
			[ 360062976, 1 ], [ 387317700, 1 ], [ 394680000, 1 ], [ 402902500, 1 ],
			[ 464143680, 1 ], [ 464257024, 1 ], [ 469945476, 2 ], [ 483483000, 1 ],
			[ 485760000, 1 ], [ 502078500, 1 ], [ 503513010, 1 ], [ 504627200, 1 ],
			[ 517899096, 1 ], [ 522161640, 1 ], [ 551675124, 1 ], [ 557865000, 1 ],
			[ 655360000, 1 ], [ 805805000, 1 ], [ 1021620600, 1 ] ]</pre>

Vipul: Created page with "{{group-specific information| information type = linear representation theory| group = Conway group:Co0| connective = of}}"

2012-05-21T19:26:42Z

Created page with "{{group-specific information| information type = linear representation theory| group = Conway group:Co0| connective = of}}"

New page

{{group-specific information|
information type = linear representation theory|
group = Conway group:Co0|
connective = of}}

Linear representation theory of Conway group:Co0 - Revision history

Vipul at 19:35, 21 May 2012

Vipul: Created page with "{{group-specific information| information type = linear representation theory| group = Conway group:Co0| connective = of}}"