Standard Young tableaux, representations and Jucys-Murphy elements

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

Last update: 27 February 2013

Standard Young tableaux, representations and Jucys-Murphy elements

In this section we review the generalization of standard Young tableaux in [Ram0401326] which is used to construct representations of the affine Hecke algebras H,1,n. Then we show how this theory can be transported to provide combinatorial constructions of simple modules for the cyclotomic Hecke algebras Hr,1,n (u1,,ur;q) and Hr,p,n (x0,,xd-1;q) . This approach shows that Jucys-Murphy type elements in the cyclotomic Hecke algebras arise naturally as images of the elements Xεi in the affine Hecke algebra. The standard Jucys-Murphy type theorems then follow almost immediately from standard affine Hecke algebra facts.

3.1 Skew shapes and standard tableaux

A partition λ is a collection of n boxes in a corner. We shall conform to the conventions in [Mac1995] and assume that gravity goes up and to the left.

Any partition λ can be identified with the sequence λ=(λ1λ2) where λi is the number of boxes in row i of λ. The rows and columns are numbered in the same way as for matrices. In the example above we have λ=(553311). If λ and μ are partitions such that μiλi for all i we write μλ. The skew shape λ/μ consists of all boxes of λ which are not in μ. Any skew shape is a union of connected components. Number the boxes of each skew shape λ/μ along major diagonals from southwest to northeast and

writeboxito indicate the box numbered i.

Let λ/μ be a skew shape with n boxes. A standard tableau of shape λ/μ is a filling of the boxes in the skew shape λ/μ with the numbers 1,,n such that the numbers increase from left to right in each row and from top to bottom down each column.

3.2 Placed skew shapes

Let +i[0,2π/ln(q2)) = { a+bia,0 b2π/ln(q2) } . If q is a positive real number then the function

+i[0,2π/ln(q2)) * x q2x=eln(q2)x

is a bijection. The elements of [0,1)+i [0,2π/ln(q2)) index the -cosets in +i[0,2π/ln(q2)).

A placed skew shape is a pair (c,λ/μ) consisting of a skew shape λ/μ and a content function

c: {boxes ofλ/μ} +i[0,2π/ln(q2)) such that (3.3) c(boxi)-c (boxi)0 ifi<jandc (boxj)-c (boxi), c(boxj)=c (boxi)+1, ifboxiand boxjare on adjacent diagonals, and c(boxi)=c (boxj), ifboxiand boxjare on the same diagonal.

This is a generalization of the usual notion of the content of a box in a partition (see [Mac1995] I §1 Ex. 3).

Suppose that (c,λ/μ) is a placed skew shape such that c takes values in . One can visualize (c,λ/μ) by placing λ/μ on a piece of infinite graph paper where the diagonals of the graph paper are indexed consecutively (with elements of ) from southeast to northwest. The content of a box b is the index c(b) of the diagonal that b is on. In the general case, when c takes values in +i[0,2π/ln(q2)), one imagines a book where each page is a sheet of infinite graph paper with the diagonals indexed consecutively (with elements of ) from southeast to northwest. The pages are numbered by values β[0,1)+i[0,2π/ln(q2)) and there is a skew shape λ(β)/μ(β) placed on page β. The skew shape λ/μ is a union of the disjoint skew shapes on each page,

λ/μβ ( λ(β)/ μ(β) ) ,β [0,1)+i [0,2π/ln(q2)) , (3.4)

and the content function is given by

c(b)= (page number of the page containingb)+ (index of the diagonal containingb). (3.5)

for a box bλ/μ.

3.6 Example

The following diagrams illustrate standard tableaux and the numbering of boxes in a skew shape λ/μ.

10 12 13 14 6 8 11 5 7 9 4 2 3 1 3 4 9 12 1 5 10 7 13 14 2 6 8 11 λ/μwith boxes numbered A standard tableauLof shape λ/μ

The following picture shows the contents of the boxes in the placed skew shape (c,λ/μ) such that the sequence

( c(box1),, c(boxn) ) is ( -7,-6,-5,-2,0, 1,1,2,2,3,3,4, 5,6 ) . 3 4 5 6 1 2 3 0 1 2 -2 -6 -5 -7 Contents of the boxes of (c,λ/μ)

The following picture shows the contents of the boxes in the placed skew shape (c,λ/μ) such that

( c(box1),, c(boxn) ) = ( -7,-6,-5,-32, 12,32,32, 52,52,72, 72,92,112, 132 ) . 72 92 112 132 -6 -5 32 52 72 -7 12 32 52 -32 0 12

This “book” has two pages, with page numbers 0 and 12.

3.7 Calibrated H,1,n-modules

A finite dimensional H,1,n-modules M is calibrated if it has a basis {vt} such that for each λL and each vt in the basis

Xλvt=t(Xλ) vt,for somet(Xλ) .

This is the class of representations of the affine Hecke algebra for which there is a good theory of Young tableaux [Ram2003].

The following theorem classifies and constructs all irreducible calibrated representations of the affine Hecke algebra H,1,n. The construction is a direct generalization of A. Young’s classical “seminormal construction” of the irreducible representations of the symmetric group [You1931]. Young’s construction was generalized to Iwahori-Hecke algebras of type A by Hoefsmit [Hoe1974] and Wenzl [Wen1988] independently, to Iwahori-Hecke algebras of types B and D by Hoefsmit [Hoe1974] and to cyclotomic Hecke algebras by Ariki and Koike [AKo1994]. In (3.7) and (3.11) below we show how all of these earlier generalizations of Young’s construction can be obtained from Theorem 3.8. Some parts of Theorem 3.8 are originally due to I. Cherednik, and are stated in [Che1987, §3].

Theorem 3.8. ([Ram1997, Theorem 4.1]) Let (c,λ/μ) be a placed skew shape with n boxes. Define an action of H,1,n on the vector space

H(c,λ/μ)= -span { vLL is a standard tableau of shapeλ/μ }

by the formulas

XεivL = q2c(L(i)) vL, TivL = (Ti)LL vL+ ( q-1+ (Ti)LL ) vsiL,

where siL is the same as L except that the entries i-1 and i are interchanged,

(Ti)LL = q-q-1 1- q2 ( c(L(i-1))- c(L(i)) ) , vsiL = 0,ifsiL is not a standard tableau,

and L(i) denotes the box of L containing the entry i.

  1. H(c,λ/μ) is a calibrated irreducible H,1,n-module.
  2. The modules H(c,λ/μ) are non-isomorphic.
  3. Every irreducible calibrated H,1,n-module is isomorphic to H(c,λ/μ) for some placed skew shape (c,λ/μ).

Remark 3.9. All of the irreducible modules for the affine Hecke algebra have been classified and constructed by Kazhdan and Lusztig [KLu0862716]. The construction in [KLu0862716] is geometric and noncombinatorial. It is nontrivial (but not very difficult) to relate the construction of Theorem 3.8 and the classification in [KLu0862716].

3.10 Calibrated H,p,n-modules

A finite dimensional H,p,n module M is calibrated if it has a basis {vt} such that for each λLp and each vt in the basis

Xλvt=t(Xλ) vt,for somet (Xλ).

Let us show how Theorem 3.8, Theorem 2.2 and Theorem A.13 provide explicit constructions of simple calibrated H,p,n-modules. The resulting construction is a generalization of the construction of Hr,p,n (x0,,xd-1;q) -modules given by Ariki [Ari1995] (as amplified and applied in [HRa1998]). Comparing the following machinations with those in [HRa1998, §3] (where more pictures are given) will be helpful.

The (/p)-action on H,1,n induces an action of /p on the simple H,1,n-modules, as in (A.1) of the Appendix, and this action takes calibrated modules to calibrated modules since the (/p)-action on H,1,n preserves the subalgebra [X]. The /p action on simple calibrated modules can be described combinatorially as follows.

If (c,λ/μ) is a placed skew shape with n boxes and g/p define

g(c,λ/μ)= (c-iα/p,λ/μ), whereα=2π/ ln(q2), (3.11)

and c-iα/p denotes the content function defined by (c-iα/p)(b)= c(b)-iα/p, for all boxes bλ/μ. To make this definition we are identifying the set [0,α) with /α. One can imagine the placed skew shape as a book with pages numbered by values β[0,1)+i (/α) and a skew shape λ(β)/ μ(β) on each page. The action of

gcyclically permutes the pages numbered β+i(k/p)α, 0k<p.

If L is a standard tableau of shape (c,λ/μ) let gL denote the same filling of λ/μ as L but viewed as a standard tableaux of shape g(c,λ/μ).

Let gH(c,λ/μ) be the H,1,n-module H(c,λ/μ) except twisted by the automorphism g, see (A.1) in the Appendix. It follows from the formulas in Theorem 3.8 that the map

ϕ: gH(c,λ/μ) Hg(c,λ/μ) vL vgL (3.12)

is an H,1,n-module isomorphism. Indeed, since g-1(Tj)=Tj and (Tj)gL,gL=(Tj)LL,

ϕ(TjvL) = Tjϕ(vL), and φ(XεjvL) = ϕ(g-1(Xεj)vL) = ϕ ( ξ-1 XεjvL ) = q2 ( c(L(j)) -iα/p ) vgL = Xεj vgL = Xεjϕ(vL),

where denotes the H,1,n-action on gH(c,λ/μ) as in (A.1). Identify gH(c,λ/μ) with Hg(c,λ/μ) via the isomorphism in (3.12).

Let (c,λ/μ) be a placed skew shape with n boxes and let K(c,λ/μ) be the stabilizer of (c,λ/μ) under the action of /p. The cyclic group K(c,λ/μ) is a realization of the inertia group of H(c,λ/μ) and

K(c,λ/μ)= { (gκ): H(c,λ/μ) H(c,λ/μ) 0 K(c,λ/μ) -1 } ,

where κ is the smallest integer between 1 and p such that gκ (c,λ/μ)= (c,λ/μ) and K(c,λ/μ) is the order of K(c,λ/μ). The elements of K(c,λ/μ) are all H,p,n-module isomorphisms. Since K(c,λ/μ) is a cyclic group the irreducible K(c,λ/μ)-modules are all one-dimensional and the characters of these modules are given explicitly by

ηj: K(c,λ/μ) gκ ξjκ, 0j K(c,λ/μ) -1,

since ξκ is a primitive K(c,λ/μ)-th root of unity. The element

pj= =0 K(c,λ/μ)-1 ξ-jκ gκ. (3.13)

is the minimal idempotent of the group algebra K(c,λ/μ) corresponding to the irreducible character ηj. It follows (from a standard double centralizer result, [Bou1958]) that, as an ( H,p,n, K(c,λ/μ) ) -bimodule,

H(c,λ/μ) j=0 K(c,λ/μ)-1 H(c,λ/μ,j) K(j),where H(c,λ/μ,j)= pjH(c,λ/μ), (3.14)

and K(j) is the irreducible K(c,λ/μ)-module with character ηj. The following theorem now follows from Theorem A.13 of the Appendix.

Theorem 3.15 Let (c,λ/μ) be a placed skew shape with n boxes and let H(c,λ/μ) be the simple calibrated H,1,n-module constructed in Theorem 3.8. Let K(c,λ/μ) be the inertia group of H(c,λ/μ) corresponding to the action of /p on H,1,n defined by (A.1). If pj is the minimal idempotent of K(c,λ/μ) given by (3.13) then

H(c,λ/μ,j)= pjH(c,λ/μ)

is a simple calibrated H,p,n-module.

Theorem 3.15 provides a generalization of the construction of the Hr,p,n-modules which was given by Ariki [Ari1995] and extended and applied in [HRa1998].

3.16 Simple Hr,1,n(u1,,ur;q)-modules

Many (usually all) of the simple Hr,1,n(u1,,ur;q)-modules can be constructed with Theorems 2.8 and 3.8.

If λ/μ is a skew shape define

NW(λ/μ)= {northwest corner boxes ofλ/μ},

so that NW(λ/μ) is the set of boxes bλ/μ such that there is no box of λ/μ immediately above or immediately to the left of b.

Theorem 3.17. Fix u1,,ur* and let (c,λ/μ) be a placed skew shape with n boxes. If

{ q2c(b) bNW(λ/μ) } {u1,,ur}

then the H,1,n-module H(c,λ/μ) is a simple Hr,1,n (u1,,ur;q) -module (via Theorem 2.8).


Let (c,λ/μ) be a placed skew shape with n boxes and let H(c,λ/μ) be the simple H,1,n-module of Theorem 3.8. Let ρ(c,λ/μ): H,1,nEnd (H(c,λ/μ)) be the representation corresponding to H(c,λ/μ). By the formulas in Theorem 3.8 the matrix ρ(Xε1) is diagonal with eigenvalues q2c(L(1)), for L(c,λ/μ), where (c,λ/μ) is the set of standard tableaux of shape λ/μ.

The boxes L(1), L(c,λ/μ) are exactly the northwest corner boxes of λ/μ and so the minimal polynomial of ρ(Xε1) is

p(t)= bNW(λ/μ) (t-q2c(b)).

Thus the condition { q2c(b) bNW(λ/μ) } {u1,,ur} is exactly what is needed for H(c,λ/μ) to be an Hr,1,n(u1,,ur;q)-module via Theorem 2.8.

Theorem 3.18. If the cyclotomic Hecke algebra Hr,1,n(u1,u2,,ur;q) is semisimple, then its simple modules are the modules H(c,λ) constructed in Theorem 3.17, where λ=(λ(1),,λ(r)) is an r-tuple of partitions with a total of n boxes and c is the content function determined by

q2c(b)=ui, ifb is the northwest corner box of λ(i).


The equations q2c(b)=ui, for bNW(λ(i)), determine the values c(b) for all boxes bλ and thus the pair (c,λ) defines a placed skew shape. By a Theorem of Ariki [Ari1994], Hr,1,n(u1,,ur;q) is semisimple if and only if

[n]q!0and uiuj-1 {1,q2,,q2n},

where [n]q!=[n]q [n-1]q [2]q [1]q and [k]q= (qk-q-k)/ (q-q-1). These conditions guarantee that (c,λ) is a placed skew shape and that the H,1,n-module H(c,λ) defined in Theorem 3.8 is well defined and irreducible. The reduction in Theorem 2.8 makes H(c,λ) into a simple Hr,1,n(u1,,ur;q)-module and a count of standard tableaux (using binomial coefficients and the classical identity for the symmetric group case) shows that

λ=(λ(1),,λ(r)) dim(H(c,λ))2= rnn!=dim (Hr,1,n),

where the sum is over all r-tuples λ=(λ(1),,λ(r)) with a total of n boxes. Thus the H(c,λ) are a complete set of simple Hr,1,n(u1,,ur;q)-modules.

Theorem 3.18 demonstrates that the construction of simple modules for Hr,1,n by Ariki and Koike [AKo1994, Theorem 3.7], for H2,1,n(p,-p-1;q) by Hoefsmit [Hoe1974], for H1,1,n(1;q) by Hoefsmit [Hoe1974] and Wenzl [Wen1988] (independently), and for Sn=H1,1,n(1;1) and WBn=H2,1,n(1,1;1) by Young [You1929,You1931], are all special cases of Theorem 3.8.

3.19 Jucys-Murphy elements in cyclotomic Hecke algebras

The following result is well known, but we give a new proof which shows that the cyclotomic Hecke algebra analogues of the Jucys-Murphy elements which have appeared in the literature (see [BMM1993], [Ram1997], [DJM1995] and the references there) come naturally from the affine Hecke algebra H,1,n.

Corollary 3.20. Let Hr,1,n(u1,,ur;q) and Hr,p,n(x0,,xd-1;q) be the cyclotomic Hecke algebras defined in (1.1) and (1.2).

  1. The elements Mi=TiT2 T1T2Ti, 1in, generate a commutative subalgebra of Hr,1,n(u1,,ur;q).
  2. If Hr,1,n(u1,,ur;q) is semisimple then every simple Hr,1,n(u1,,ur;q)-module has a basis of simultaneous eigenvectors of the elements Mi.
  3. The elements M1p=a0, andMi M1-1=ai a3a2a1a3 ai,2in, generate a commutative subalgebra of Hr,p,n(x0,,xd-1;q).
  4. If Hr,p,n(x0,,xd-1;q) is semisimple then every simple Hr,p,n(x0,,xd-1;q)-module has a basis of simultaneous eigenvectors of the elements M1p and MiM1-1, 2in.


(a) The elements Xεi, 1in, generate the subalgebra [X]H,1,n. Inductive use of the relation (1.4) shows that Mi=Φ(Xεi), where Φ:H,1,n Hr,1,n (u1,,ur;q) is the homomorphism in Proposition 2.5. Thus the subalgebra of Hr,1,n(u1,,ur;q) generated by the element Mi is the image of [X] under the homomorphism Φ.

(b) is an immediate consequence of Theorem 3.18, the construction described in Theorem 3.8, and the fact that Mi=Φ(Xεi).

(c)The elements Xpε1 and Xεi-ε1, 2in, generate the subalgebra [XLp]H,p,n and the images of these elements under the homomorphism Φp: H,p,n Hr,p,n (x0,,xd-1;q) are the elements M1p and MiM1-1.

The proof of part (d) uses the construction described in Theorem 3.15 and is analogous to the proof of part (b).

(3.21) The center of Hr,1,n(u1,,ur;q)

It is an immediate consequence of (1.12) and the proof of Corollary 3.20 that

[ M1,, Mn ] Sn Z ( Hr,1,n (u1,,ur;q) ) ,

where Mi=TiTi-1 T2T1T2 Ti-1Ti. The following proposition shows that this inclusion is an equality.

Proposition 3.22. If the cyclotomic Hecke algebra Hr,1,n(u1,,ur;q) is semisimple then its center

Z ( Hr,1,n (u1,,ur;q) ) = [ M1,, Mn ] Sn ,

where Mi=TiTi-1 T2T1T2 Ti-1Ti and [ M1,, Mn ] Sn is the ring of symmetric polynomials in M1,,Mn.


By (1.12) Z(H,1,n)= [X]Sn. Thus, since Mi=Φ(Xεi), where Φ:H,1,n Hr,1,n (u1,,ur;q) is the surjective homomorphism of Proposition 2.5, it follows that

[M1,,Mn]Sn=Φ ([X]Sn)=Φ (Z(H,1,n))Z ( Hr,1,n (u1,,ur;q) ) .

For the reverse inclusion we need to show that the action of the elements [M1,,Mn]Sn distinguishes the simple Hr,1,n(u1,,ur;q)-modules.

Let λ=(λ(1),,λ(r)) be an r-tuple of partitions with a total of n boxes and let H(c,λ) be the corresponding simple Hr,1,n(u1,,ur;q)-module as constructed by Theorem 3.18 (and Theorem 3.8). If L is a standard tableau of shape λ then M1,,Mn act on vL by the multiset of values ( q2c(L(1)) ,, q2c(L(n)) ) . The elementary symmetric functions ei(M1,,Mn) act on H(c,λ) by the values ai=ei ( q2c(L(1)) ,, q2c(L(n)) ) . Note that ai does not depend on the choice of the standard tableau L (since ei(M1,,Mn)Z(Hr,1,n)). We show that the simple module H(c,λ) is determined by the values a1,,an. This shows that the simple modules are distinguished by the elements of [M1,,Mn]Sn.

Let us explain how the values a1,,an determine the placed skew shape (c,λ). There is a unique (unordered) multiset of values b1,,bn such that ai=ei(b1,,bn) for all 1in. The bi are determined by the equation

(t-b1) (t-bn)= tn-a1tn-1 +a2tn-2- a3tn-3+ ±an.

From the previous paragraph, the multiset {b1,,bn} must be the same as the multiset { q2c(L(1)) ,, q2c(L(n)) } . In this way the values a1,,an determine the multiset S={c(L(1)),,c(L(n))}.

The multiset S is a disjoint union S=S1Sr of multisets such that each Si is in a single -coset of +i[0,2π/ln(q2))i (see (3.4)). These cosets are determined by the values u1,,ur and thus there is a one-to-one correspondence between the ui and the multisets Si. Then λ(i)=Card(Si), the nonempty diagonals of λ(i) are determined by the values in Si, and the lengths of the diagonals of λ(i) are the multiplicities of the values of the elements of Si. This information completely determines λ(i) for each i. Thus (c,λ) is determined by the values a1,,an.

Remark 3.23. In the language of affine Hecke algebra representations (see [Ram2003]) the proof of Proposition 3.22 shows that (when Hr,1,n(u1,,ur;q) is semisimple) the simple Hr,1,n(u1,,ur;q)-modules H(c,λ) all have different central characters (as H,1,n modules).

Remark 3.24. The elements M1p and MiM1-1 from Corollary 3.20 cannot be used to obtain a direct analogue of Proposition 3.22 for Hr,p,n(x0,,xd-1;q). This is because all of the H,p,n-modules Vj appearing in the decomposition (3.14) will have the same central character. However, when n is odd, the natural analogue of Proposition 3.22 does hold for Iwahori-Hecke algebras of type Dn, HDn(q)=H2,2,n(1;q). In that case, every simple H2,2,n(1;q). has trivial inertia group and the decomposition in (3.14) has only one summand.

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

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

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