Hello,
> On Mon, Aug 18, 2014 at 09:15:26AM -0700, Biswajit Ransingh wrote:
> > W=WeylGroup(['E',8])
> >
> > R = RootSystem(['E',8]).root_lattice()
> >
> > alpha = R.simple_roots();alpha
> >
> > [w for w in W if w.action(alpha[1])==(alpha[1]+alpha[2])]
> >
> > Output shows : gap: cannot extend the workspace any more!
As explained, you cannot construct E8 in memory in GAP, it is too large.
However what you sem to want is easy both theoretically and in pratice.
What you seem to want is the coset of the fixator of the first root which
brings it to alpha1+alpha2.
I guess the indexing of simple roots is different in Chevie and Sage since
in Chevie alpha1+alpha2 is not a root but alpha1+alpha3 is.
gap> W:=CoxeterGroup("E",8);
CoxeterGroup("E",8)
gap> W.roots[10];
[ 0, 1, 0, 1, 0, 0, 0, 0 ]
gap> RepresentativeOperation(W,1,10);
( 1, 10, 16,123, 23, 4)( 2, 11,137)( 3,143,124,121,130,136)( 5, 25, 19)
( 9,129)( 12, 30, 32, 24, 18, 44)( 13, 33, 27)( 17,122,131)( 20, 38, 40, 31,
26, 51)( 21, 41, 35)( 28, 46, 49, 39, 34, 58)( 29, 50, 43)( 36, 54, 56, 47,
42, 65)( 48, 57)( 52, 69, 63)( 55, 64)( 59, 75, 70)( 61, 71)( 62, 72)
( 66, 80, 76)( 67, 81, 77)( 68, 78)( 73, 86, 83)( 74, 84)( 79, 91, 88)
( 82, 85, 93)( 87, 90, 98)( 89, 97)( 92, 95,102)( 94,101)( 96,100,105)
( 99,104)(103,107)(106,115,110,109,112,114)(111,113,116)(125,145,139)
(132,150,152,144,138,164)(133,153,147)(140,158,160,151,146,171)(141,161,155)
(148,166,169,159,154,178)(149,170,163)(156,174,176,167,162,185)(168,177)
(172,189,183)(175,184)(179,195,190)(181,191)(182,192)(186,200,196)
(187,201,197)(188,198)(193,206,203)(194,204)(199,211,208)(202,205,213)
(207,210,218)(209,217)(212,215,222)(214,221)(216,220,225)(219,224)(223,227)
(226,235,230,229,232,234)(231,233,236)
The above element sends the root no1 to root no10 which is alpha1+alha3
gap> l:=Filtered([1..W.N],i->W.1^Reflection(W,i)=W.1);
[ 1, 2, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 18, 20, 21, 22, 26, 28, 29, 34,
36, 42, 44, 51, 57, 58, 63, 64, 65, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81,
82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 99, 100, 103, 106,
114, 115, 116, 117, 118, 119, 120 ]
gap> R:=ReflectionSubgroup(W,l);
ReflectionSubgroup(CoxeterGroup("E",8), [ 1, 2, 4, 5, 6, 7, 8, 44 ])
R is the subgroup generated by the roots orthogonal to alpha1 and s_alpha1
gap> PrintDiagram(R);
A1 1
E7 44
|
8 - 7 - 6 - 5 - 4 - 2
It is A1 x E7
brk> gap> N:=Centralizer(W,W.1);
Subgroup( CoxeterGroup("E",8),
[ ( 8,239)( 15,238)( 22,237)( 29,236)( 36,235)( 42,234)( 43,233)( 47,232)
( 50,231)( 54,230)( 56,229)( 60,228)( 62,227)( 65,226)( 67,225)( 68,224)
( 72,223)( 73,222)( 74,221)( 77,220)( 78,219)( 79,218)( 81,216)( 83,215)
( 84,214)( 86,212)( 87,211)( 88,210)( 90,208)( 91,207)( 92,206)( 94,204)
( 95,203)( 96,201)( 98,199)( 99,198)(100,197)(101,194)(102,193)(103,192)
(104,188)(105,187)(106,185)(107,182)(108,180)(109,176)(110,174)(111,170)
(112,167)(113,163)(114,162)(115,156)(116,149)(117,142)(118,135)(119,128)
(120,240), ( 7, 15)( 8,128)( 14, 22)( 21, 29)( 28, 36)( 34, 42)( 35, 43)
( 39, 47)( 41, 50)( 46, 54)( 49, 56)( 53, 60)( 55, 62)( 58, 65)( 59, 67)
( 61, 68)( 64, 72)( 66, 73)( 70, 77)( 71, 78)( 75, 81)( 76, 83)( 80, 86)
( 82, 87)( 85, 90)( 89, 94)( 93, 98)( 97,101)(119,120)(127,135)(134,142)
(141,149)(148,156)(154,162)(155,163)(159,167)(161,170)(166,174)(169,176)
(173,180)(175,182)(178,185)(179,187)(181,188)(184,192)(186,193)(190,197)
(191,198)(195,201)(196,203)(200,206)(202,207)(205,210)(209,214)(213,218)
(217,221)(239,240), ( 7,238)( 8,120)( 14,237)( 21,236)( 28,235)( 34,234)
( 35,233)( 39,232)( 41,231)( 46,230)( 49,229)( 53,228)( 55,227)( 58,226)
( 59,225)( 61,224)( 64,223)( 66,222)( 70,220)( 71,219)( 74,217)( 75,216)
( 76,215)( 79,213)( 80,212)( 82,211)( 84,209)( 85,208)( 88,205)( 89,204)
( 91,202)( 92,200)( 93,199)( 95,196)( 96,195)( 97,194)( 99,191)(100,190)
(102,186)(103,184)(104,181)(105,179)(106,178)(107,175)(108,173)(109,169)
(110,166)(111,161)(112,159)(113,155)(114,154)(115,148)(116,141)(117,134)
(118,127)(119,239)(128,240), ( 6, 14)( 7,127)( 8, 15)( 13, 21)( 20, 28)
( 26, 34)( 27, 35)( 31, 39)( 33, 41)( 38, 46)( 40, 49)( 45, 53)( 48, 55)
( 51, 58)( 52, 59)( 57, 64)( 63, 70)( 68, 74)( 69, 75)( 73, 79)( 78, 84)
( 83, 88)( 86, 91)( 87, 92)( 90, 95)( 94, 99)( 98,102)(101,104)(118,119)
(126,134)(128,135)(133,141)(140,148)(146,154)(147,155)(151,159)(153,161)
(158,166)(160,169)(165,173)(168,175)(171,178)(172,179)(177,184)(183,190)
(188,194)(189,195)(193,199)(198,204)(203,208)(206,211)(207,212)(210,215)
(214,219)(218,222)(221,224)(238,239), ( 5, 13)( 6,126)( 7, 14)( 12, 20)
( 15, 22)( 18, 26)( 19, 27)( 24, 31)( 25, 33)( 30, 38)( 32, 40)( 37, 45)
( 44, 51)( 55, 61)( 59, 66)( 62, 68)( 64, 71)( 67, 73)( 70, 76)( 72, 78)
( 75, 80)( 77, 83)( 81, 86)( 92, 96)( 95,100)( 99,103)(102,105)(104,107)
(117,118)(125,133)(127,134)(132,140)(135,142)(138,146)(139,147)(144,151)
(145,153)(150,158)(152,160)(157,165)(164,171)(175,181)(179,186)(182,188)
(184,191)(187,193)(190,196)(192,198)(195,200)(197,203)(201,206)(212,216)
(215,220)(219,223)(222,225)(224,227)(237,238),
( 5, 21)( 6,127)( 7,126)( 8, 22)( 12, 28)( 14,134)( 18, 34)( 19, 35)
( 24, 39)( 25, 41)( 30, 46)( 32, 49)( 37, 53)( 44, 58)( 48, 61)( 52, 66)
( 57, 71)( 62, 74)( 63, 76)( 67, 79)( 69, 80)( 72, 84)( 77, 88)( 81, 91)
( 87, 96)( 90,100)( 94,103)( 98,105)(101,107)(117,119)(125,141)(128,142)
(132,148)(138,154)(139,155)(144,159)(145,161)(150,166)(152,169)(157,173)
(164,178)(168,181)(172,186)(177,191)(182,194)(183,196)(187,199)(189,200)
(192,204)(197,208)(201,211)(207,216)(210,220)(214,223)(218,225)(221,227)
(237,239), ( 5, 29)( 6,135)( 8,134)( 12, 36)( 14,128)( 15,126)( 18, 42)
( 19, 43)( 22,142)( 24, 47)( 25, 50)( 30, 54)( 32, 56)( 37, 60)( 44, 65)
( 48, 68)( 52, 73)( 55, 74)( 57, 78)( 59, 79)( 63, 83)( 64, 84)( 69, 86)
( 70, 88)( 75, 91)( 82, 96)( 85,100)( 89,103)( 93,105)( 97,107)(117,120)
(125,149)(132,156)(138,162)(139,163)(144,167)(145,170)(150,174)(152,176)
(157,180)(164,185)(168,188)(172,193)(175,194)(177,198)(179,199)(183,203)
(184,204)(189,206)(190,208)(195,211)(202,216)(205,220)(209,223)(213,225)
(217,227)(237,240), ( 4, 12)( 5,125)( 6, 13)( 10, 18)( 11, 19)( 14, 21)
( 16, 24)( 17, 25)( 22, 29)( 23, 30)( 40, 48)( 45, 52)( 49, 55)( 51, 57)
( 53, 59)( 56, 62)( 58, 64)( 60, 67)( 65, 72)( 76, 82)( 80, 85)( 83, 87)
( 86, 90)( 88, 92)( 91, 95)(103,106)(105,108)(107,109)(116,117)(124,132)
(126,133)(130,138)(131,139)(134,141)(136,144)(137,145)(142,149)(143,150)
(160,168)(165,172)(169,175)(171,177)(173,179)(176,182)(178,184)(180,187)
(185,192)(196,202)(200,205)(203,207)(206,210)(208,212)(211,215)(223,226)
(225,228)(227,229)(236,237), ( 4, 20)( 5,126)( 6,125)( 7, 21)( 10, 26)
( 11, 27)( 13,133)( 15, 29)( 16, 31)( 17, 33)( 23, 38)( 32, 48)( 37, 52)
( 44, 57)( 49, 61)( 53, 66)( 56, 68)( 58, 71)( 60, 73)( 65, 78)( 70, 82)
( 75, 85)( 77, 87)( 81, 90)( 88, 96)( 91,100)( 99,106)(102,108)(104,109)
(116,118)(124,140)(127,141)(130,146)(131,147)(135,149)(136,151)(137,153)
(143,158)(152,168)(157,172)(164,177)(169,181)(173,186)(176,188)(178,191)
(180,193)(185,198)(190,202)(195,205)(197,207)(201,210)(208,216)(211,220)
(219,226)(222,228)(224,229)(236,238),
( 3, 98,108, 93)( 4,189)( 5,135,223,127)( 6,126)( 7,125, 15,103)
( 8,106,128,226)( 9,101,109, 97)( 10,183)( 11, 43,110, 35)
( 12,201,220,195)( 13,142,219,134)( 14,133, 22, 99)( 16, 47,111, 39)
( 17, 50,112, 41)( 18,197,216,190)( 19, 27)( 20,206,215,200)
( 21,149,209, 94)( 23, 54,113, 46)( 24, 31)( 25, 33)( 26,203,212,196)
( 28,210,205, 36)( 29, 89,214,141)( 30, 38)( 32,157)( 34,207,202, 42)
( 37,152)( 40,165)( 44, 78,117, 71)( 45,160)( 48,180,199,173)
( 49,172, 56, 74)( 51, 72,118, 64)( 52,176,194,169)( 53,168, 60, 79)
( 55,187,186, 68)( 58, 65,120,119)( 59,182,181, 73)( 61,193,179, 62)
( 63,130)( 66,188,175, 67)( 69,124)( 70,138, 77, 96)( 75,132, 81,100)
( 76,146, 83, 92)( 80,140, 86, 95)( 82,162,154, 87)( 84,116)
( 85,156,148, 90)(102,105)(104,107)(123,218,228,213)(129,221,229,217)
(131,163,230,155)(136,167,231,159)(137,170,232,161)(139,147)
(143,174,233,166)(144,151)(145,153)(150,158)(164,198,237,191)
(171,192,238,184)(178,185,240,239)(204,236)(222,225)(224,227),
( 3,108)( 5,223)( 7, 15)( 8,128)( 9,109)( 11,110)( 12,220)( 13,219)
( 14, 22)( 16,111)( 17,112)( 18,216)( 20,215)( 21,209)( 23,113)( 26,212)
( 28,205)( 29,214)( 34,202)( 35, 43)( 36,210)( 39, 47)( 41, 50)( 42,207)
( 44,117)( 46, 54)( 48,199)( 49, 56)( 51,118)( 52,194)( 53, 60)( 55,186)
( 58,120)( 59,181)( 61,179)( 62,193)( 64, 72)( 65,119)( 66,175)( 67,188)
( 68,187)( 70, 77)( 71, 78)( 73,182)( 74,172)( 75, 81)( 76, 83)( 79,168)
( 80, 86)( 82,154)( 85,148)( 87,162)( 89,141)( 90,156)( 92,146)( 93, 98)
( 94,149)( 95,140)( 96,138)( 97,101)( 99,133)(100,132)(103,125)(106,226)
(123,228)(127,135)(129,229)(131,230)(134,142)(136,231)(137,232)(143,233)
(155,163)(159,167)(161,170)(164,237)(166,174)(169,176)(171,238)(173,180)
(178,240)(184,192)(185,239)(190,197)(191,198)(195,201)(196,203)(200,206)
(213,218)(217,221), ( 2, 10)( 3, 11)( 4,124)( 5, 12)( 9, 16)( 13, 20)
( 21, 28)( 25, 32)( 29, 36)( 30, 37)( 33, 40)( 38, 45)( 41, 49)( 46, 53)
( 50, 56)( 54, 60)( 57, 63)( 64, 70)( 71, 76)( 72, 77)( 78, 83)( 84, 88)
( 85, 89)( 90, 94)( 95, 99)(100,103)(108,110)(109,111)(115,116)(122,130)
(123,131)(125,132)(129,136)(133,140)(141,148)(145,152)(149,156)(150,157)
(153,160)(158,165)(161,169)(166,173)(170,176)(174,180)(177,183)(184,190)
(191,196)(192,197)(198,203)(204,208)(205,209)(210,214)(215,219)(220,223)
(228,230)(229,231)(235,236), ( 2, 18)( 3, 19)( 4,125)( 5,124)( 6, 20)
( 9, 24)( 12,132)( 14, 28)( 17, 32)( 22, 36)( 23, 37)( 33, 48)( 38, 52)
( 41, 55)( 46, 59)( 50, 62)( 51, 63)( 54, 67)( 58, 70)( 65, 77)( 71, 82)
( 78, 87)( 80, 89)( 84, 92)( 86, 94)( 91, 99)(100,106)(105,110)(107,111)
(115,117)(122,138)(123,139)(126,140)(129,144)(134,148)(137,152)(142,156)
(143,157)(153,168)(158,172)(161,175)(166,179)(170,182)(171,183)(174,187)
(178,190)(185,197)(191,202)(198,207)(200,209)(204,212)(206,214)(211,219)
(220,226)(225,230)(227,231)(235,237), ( 2, 75)( 4,184)( 6, 76)( 7,183)
( 8, 77)( 11,179)( 12,178)( 13, 82)( 16,175)( 19,173)( 22, 88)( 24,169)
( 26, 89)( 28,164)( 29, 92)( 32,159)( 33, 93)( 35,157)( 37,155)( 38, 97)
( 39,152)( 42, 99)( 44,148)( 49,144)( 50,102)( 53,139)( 54,104)( 55,136)
( 58,132)( 59,131)( 63,127)( 64,124)( 68,110)( 70,190)( 73,111)( 78,114)
( 86,116)( 90,117)(100,119)(122,195)(126,196)(128,197)(133,202)(142,208)
(146,209)(149,212)(153,213)(158,217)(162,219)(170,222)(174,224)(188,230)
(193,231)(198,234)(206,236)(210,237)(220,239),
( 2,122)( 4, 10)( 11, 17)( 12, 18)( 16, 23)( 19, 25)( 20, 26)( 24, 30)
( 27, 33)( 28, 34)( 31, 38)( 35, 41)( 36, 42)( 39, 46)( 43, 50)( 47, 54)
( 63, 69)( 70, 75)( 76, 80)( 77, 81)( 82, 85)( 83, 86)( 87, 90)( 88, 91)
( 92, 95)( 96,100)(110,112)(111,113)(114,115)(124,130)(131,137)(132,138)
(136,143)(139,145)(140,146)(144,150)(147,153)(148,154)(151,158)(155,161)
(156,162)(159,166)(163,170)(167,174)(183,189)(190,195)(196,200)(197,201)
(202,205)(203,206)(207,210)(208,211)(212,215)(216,220)(230,232)(231,233)
(234,235), ( 2,216)( 4,103)( 6,215)( 11,105)( 12,106)( 13,211)( 14,210)
( 16,107)( 19,108)( 21,206)( 22,205)( 24,109)( 26,204)( 29,200)( 32,112)
( 33,199)( 34,198)( 37,113)( 38,194)( 41,193)( 42,191)( 44,115)( 46,188)
( 50,186)( 54,181)( 61,174)( 63,118)( 66,170)( 68,166)( 70,119)( 71,162)
( 73,161)( 74,158)( 77,120)( 78,154)( 79,153)( 80,149)( 84,146)( 85,142)
( 86,141)( 90,134)( 91,133)( 95,126)( 96,122)(100,220)(124,223)(131,225)
(132,226)(136,227)(139,228)(144,229)(152,232)(157,233)(164,235)(183,238)
(190,239)(197,240), ( 1,121)( 3, 9)( 11, 16)( 17, 23)( 19, 24)( 25, 30)
( 27, 31)( 32, 37)( 33, 38)( 35, 39)( 40, 45)( 41, 46)( 43, 47)( 48, 52)
( 49, 53)( 50, 54)( 55, 59)( 56, 60)( 61, 66)( 62, 67)( 68, 73)( 74, 79)
( 93, 97)( 98,101)(102,104)(105,107)(108,109)(110,111)(112,113)(123,129)
(131,136)(137,143)(139,144)(145,150)(147,151)(152,157)(153,158)(155,159)
(160,165)(161,166)(163,167)(168,172)(169,173)(170,174)(175,179)(176,180)
(181,186)(182,187)(188,193)(194,199)(213,217)(218,221)(222,224)(225,227)
(228,229)(230,231)(232,233), ( 3,108)( 5,223)( 9,109)( 11,110)( 12,220)
( 13,219)( 16,111)( 17,112)( 18,216)( 20,215)( 21,214)( 23,113)( 26,212)
( 28,210)( 29,209)( 34,207)( 36,205)( 42,202)( 44,117)( 48,199)( 51,118)
( 52,194)( 55,193)( 58,119)( 59,188)( 61,187)( 62,186)( 65,120)( 66,182)
( 67,181)( 68,179)( 73,175)( 74,172)( 79,168)( 82,162)( 85,156)( 87,154)
( 89,149)( 90,148)( 92,146)( 94,141)( 95,140)( 96,138)( 99,133)(100,132)
(103,125)(106,226)(123,228)(129,229)(131,230)(136,231)(137,232)(143,233)
(164,237)(171,238)(178,239)(185,240),
( 2,234,207,197, 34)( 3,143,102,173,158)( 4, 51, 20,116,127)
( 5,126,223,206, 64)( 6,103, 86,184,125)( 7,124,171,140,236)
( 8,205,138, 96,120)( 9,137,104,169,153)( 10,212,192, 80, 26)
( 11, 27,110, 98,166)( 12, 44,211, 70, 57)( 13, 99,148,235, 58)
( 14,100, 42,202,237)( 15,209,189, 84, 29)( 16, 31,111,101,161)
( 17,224, 49, 33,129)( 18,216,240,128, 85)( 21, 95,196,238,214)
( 23,222, 53, 38,123)( 25,227,193,180, 55)( 28,115,178,133,219)
( 30,225,188,176, 59)( 32,199,174, 93, 48)( 35,108, 62,186,233)
( 36, 65,195, 88, 78)( 37,194,170, 97, 52)( 39,109, 67,181,232)
( 41,136,151,231,221)( 46,131,147,230,218)( 50,217,172,157, 74)
( 54,213,168,152, 79)( 56,179,150,105, 68)( 60,175,145,107, 73)
( 61,112,159,229,187)( 66,113,155,228,182)( 69,204,149,135, 89)
( 71,106, 81,239,210)( 72,200,146,130, 92)( 75,208,198,156,185)
( 76,118, 94,141,215)( 77,154,122,114, 87)( 82,117,134,220,162)
( 90,191,226,201,119)( 91,190,177,132,164),
( 2,190,208)( 5, 95, 29)( 6,135, 80)( 7,189,206)( 8,134, 81)
( 10,184,204)( 12, 99, 36)( 13, 85,100)( 14,201,128)( 15,200,126)
( 17,179,199)( 18, 42,178)( 19,102, 43)( 20, 89,103)( 22,211, 75)
( 23,175,194)( 24,104, 47)( 25, 50,173)( 27, 93,105)( 30, 54,169)
( 31, 97,107)( 32, 56,166)( 34,164,185)( 37, 60,161)( 41,157,180)
( 44, 65,154)( 46,152,176)( 48,112, 68)( 49,150,174)( 52,113, 73)
( 53,145,170)( 55, 74,143)( 57,115, 78)( 58,138,162)( 59, 79,137)
( 63,116, 83)( 64, 84,130)( 69, 86,127)( 70, 88,122)( 82,119, 96)
( 87,117,120)( 91,195,142)(125,215,149)(132,219,156)(133,205,220)
(139,222,163)(140,209,223)(144,224,167)(147,213,225)(151,217,227)
(168,232,188)(172,233,193)(177,235,198)(183,236,203)(202,239,216)
(207,237,240) ] )
gap> Size(R);
5806080
gap> Size(N);
5806080
And it is the same as the centralizer of s_alpha1.
So you are looking for the coset of E7 described above
------------------------------------------------------------------------------
Jean MICHEL, Groupes et representations, IMJ-PRG UMR7586 tel.(33)157279144
Bureau 639 Bat. Sophie Germain Case 7012 - 75205 PARIS Cedex 13
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