The type A, root of unity case

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 14 November 2012

The type A, root of unity case

This section describes the sets (t,J) in the case of the root system of Section 5.2 when q2=e2πi/, a primitive th root of unity, >2.

Let tT. Identify t with a sequence

t=(t1,,tn) n,where t(Xεi)=ti .

For the purposes of representation theory (see Theorem 3.6) t indexes a central character (see Section 2.3) and so t can safely be replaced by any element of its W-orbit. In this case W is the symmetric group, Sn, acting by permuting the sequence t=(t1,,tn).

The cyclic group q2 of order generated by q2 acts on *. F ix a choice of a set {ξ} of coset representatives of the q2 cosets in *. Replace t with the sequence obtained by rearranging its entries to group entries in the same q2-orbit, so that

t= ( ξ1t(1),, ξkt(k) ) ,

where ξ1,,ξk are distinct representatives of the cosets in */q2 and each t(j) is a sequence of the form

t(j)= ( q2γ1,, q2γr ) ,with γ1,,γr {0,1,,-1} andγ1 γr.

As in Section 5.11 this decomposition of t into groups induces decompositions

Z(t)= j=1k Zξj(t) and P(t)= j=1k Pξj(t),

and it is sufficient to analyze the case when t consists of only one group, i.e., all the entries of t are in the same q2 coset.

Now assume that

t= ( q2γ1,, q2γn ) ,withγ1 γn,γi {0,,-1}.

Consider a page of graph paper with diagonals labeled by ,0,1,,-1,0 ,1,,-1,0,1, from southwest to northeast. For each local region (t,J), JP(t), we will construct an -periodic configuration of boxes for which the -periodic standard tableaux defined below will be in bijection with the elements of (t,J). For each 1in, the configuration will have a box numbered i, boxi, on each diagonal which is labeled γi. There are an infinite number of such diagonals containing a box numbered i, since the diagonals are labeled in an -periodic fashion, but each strip of consecutive diagonals labeled 0,1,,-1 will contain n boxes. The content of a box b (see [Mac1995, I, Section 1, Exercise 3]) is

c(b)=(the diagonal number of the box b).


Z(t) = { εj-εi i<j,γi =γj } = { εj-εi i<j,boxi andboxj are in the same diagonal }


P(t) = { ej-ei i<jand γj=γi+1 ,or i<j,γj= -1,and γi=0 } = { εj-εi i<jand boxiandboxj are in adjacent diagonals } .

We will use JP(t) to organize the relative positions of the boxes in adjacent diagonals:

Thus, t determines the number of boxes in each diagonal and K determines the relative positions of the boxes in adjacent diagonals. This information completely determines the -periodic configuration of boxes associated to the pair (t,J).

A -periodic standard tableau is an -periodic filling p of the boxes with 1,2,,n such that

  1. if i<j and boxi and boxj are in the same diagonal then p(i)< p(j),
  2. if i<j and boxi and boxj are in adjacent diagonals with boxj southwest of boxi then p(i)<p(j),
  3. if i<j and boxi and boxj are in adjacent diagonals with boxj northeast of boxi then p(i)>p(j),

where p(i) denotes the entry in boxi. An -periodic standard tableau p corresponds to a permutation in Sn via the correspondence

{standard tableaux} (t,J), p ( 12n p(1) p(2) p(n) ) .

Example. Suppose q2=e2πi/4 and

t= ( q0, q0, q0, q0, q2, q2, q2, q4, q4, q6, q6, q6, q6, q6 ) .


Z(t) = { ε2-ε1, ε3-ε1, ε4-ε1, ε3-ε2, ε4-ε2, ε4-ε3, ε6-ε5, ε7-ε5, } and P(t) = { ε5-ε1, ε5-ε2, ε5-ε3, ε5-ε4, ε6-ε1,, ε14-ε9, ε10-ε1, ε10-ε2,, ε14-ε4}.


J = { ε5-ε2, ε5-ε3, ε5-ε4, ε6-ε3, ε6-ε4, ε8-ε5, ε8-ε6, ε8-ε7, ε9-ε7, ε10-ε9, ε11-ε9, ε12-ε9, ε12-ε2, ε12-ε3, ε12-ε4, ε13-ε2, ε13-ε3, ε13-ε4, ε14-ε3, ε14-ε4 }

then the corresponding -periodic configuration of boxes and a sample -periodic standard tableau are

10 11 8 10 1 5 11 8 12 1 5 9 13 2 6 12 14 3 9 13 2 6 4 7 14 3 4 7 3 3 2 3 0 1 3 2 3 0 1 2 3 0 1 3 3 0 2 3 0 1 0 1 3 0 0 1 numbering of boxes contents of boxes 8 11 5 8 1 6 11 5 2 1 6 3 7 9 12 2 10 13 3 7 9 12 14 4 10 13 14 4 a standard tableaup

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

page history