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SUMMARY:Tushar Kanta Naik (IISER Mohali)
DTSTART:20210826T040000Z
DTEND:20210826T043000Z
DTSTAMP:20260510T201122Z
UID:GroupCraft/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GroupCraft/8
 /">Automorphisms of three canonical extensions of symmetric groups</a>\nby
  Tushar Kanta Naik (IISER Mohali) as part of World of GroupCraft\n\n\nAbst
 ract\nThe symmetric group $S_n$\, $n \\geq 2$\, has a Coxeter presentation
  with generating set $X = \\{\\tau_1\, \\dots\, \\tau_{n-1}\\}$ and defini
 ng relations\n\n(1) Involutions: $\\tau^2_i = 1$ for $1 \\leq i \\leq n 
 − 1$\;\n\n(2) Braid relations: $\\tau_i \\tau_{i+1} \\tau_i = \\tau_{i+1
 } \\tau_i \\tau_{i+1}$ for $1 \\leq i \\leq n − 2$\;\n\n(3) Far commutat
 ivity: $\\tau_i \\tau_j = \\tau_j \\tau_i$ for $|i − j|\\geq 2$.\n\nBy o
 mitting all relations of type (1) (respectively type (2) ) from the preced
 ing presentation of $S_n$\, we get presentations of the Artin braid group 
 $B_n$ (respectively the twin group $T_n$). Artin braid groups are well-stu
 died objects with nice geometrical presentations in 3-space. Apart from ma
 thematics\, they have far-reaching applications in physics and biology. Tw
 in groups can be thought of as planar analogues of braid groups. Recently 
 these groups have attracted attention from (quantum) physicists. Thus\, it
  is natural to ask about the remaining case. What kind of group do we get\
 , if we omit all relations of the third type from the above presentation o
 f $S_n$?\n\nIt follows that if we remove all relations of type (3)\, we ge
 t an odd Coxeter group whose associated Coxeter graph is a straight line o
 n $n − 1$ vertices. In this talk\, we will consider a general family of 
 odd Coxeter groups whose associated Coxeter graphs are trees\, discuss the
 ir automorphism groups and compare with the automorphism groups of Artin b
 raid groups and twin groups.\n
LOCATION:https://researchseminars.org/talk/GroupCraft/8/
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