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SUMMARY:Darij Grinberg : Shuffles in the symmetric group algebra
DTSTART:20260604T120000Z
DTEND:20260604T130000Z
DTSTAMP:20260614T225000Z
UID:indico-event-16686@indico.math.cnrs.fr
DESCRIPTION:Speakers: Darij Grinberg (Drexel University\, Philadelphie)\n\
 nEver since the famous 1992 work of Bayer and Diaconis\, it has been known
  that random shuffles of a deck of cards (with the back side up) can be mo
 delled as elements of the group algebra $\\mathbb{R}[S_n]$ of the symmetri
 c group $S_n$. This viewpoint has spawned progress in both card shuffling 
 and the representation theory of the symmetric group. In this talk\, I wil
 l focus on two projects in the latter: one focusingon the "somewhere-to-be
 low shuffles" $$t_i := (i) + (i\,i+1) + (i\,i+1\,i+2) + \\cdots + (i\,i+1
 \,\\ldots\,n) \\in \\mathbb{R}[S_n]$$for $1 \\leq i \\leq n$ (where the pa
 renthesized expressions mean cycles\; the $1$-cycle $(i)$ is the identity)
 \, and one focusing on the "$k$-random-to-random shuffles" $$R_k := \\sum
 _{1 \\leq i_1 < i_2 < \\cdots < i_k \\leq n} \\sum_{w \\in S_n \\text{ suc
 h that } w(i_1) < w(i_2) < \\cdots < w(i_k)} w \\in \\mathbb{R}[S_n]$$for 
 $0 \\leq k \\leq n$. Both families have revealed a variety of unexpected p
 roperties. For instance\, the $R_0\, R_1\, \\ldots\, R_n$ commute\, wherea
 s the $t_1\, t_2\, \\ldots\, t_n$ are simultaneously triangularizable (i.e
 .\, there is an -- explicitly describable -- basis of $\\mathbb{R}[S_n]$ o
 n which right multiplication by each $t_i$ acts as a triangular matrix). I
 n both cases\, all eigenvalues are integers and can be explicitly describe
 d and assigned to Specht modules (irreducible representations of $S_n$). M
 any of these properties furthermore generalize to the (type-A) Iwahori-Hec
 ke algebra.\nDue to the amount of results\, this talk will be an overview 
 with no proofs.\nSome of the above is joint work with Nadia Lafrenière\, 
 Sarah Brauner\, Patricia Commins and Franco Saliola.\n\nhttps://indico.mat
 h.cnrs.fr/event/16686/
LOCATION:Salle Pierre Grisvard (IHP - Bâtiment Borel)
URL:https://indico.math.cnrs.fr/event/16686/
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