Supercharacters,symmetric functions in noncommuting

variables,and related Hopf algebras

Marcelo Aguiar,Carlos Andre,Carolina Benedetti,Nantel Bergeron,

Zhi Chen,Persi Diaconis,Anders Hendrickson,Samuel Hsiao,I.Martin Isaacs,

Andrea Jedwab,Kenneth Johnson,Gizem Karaali,Aaron Lauve,Tung Le,

Stephen Lewis,Huilan Li,Kay Magaard,Eric Marberg,Jean-Christophe Novelli,

Amy Pang,Franco Saliola,Lenny Tevlin,Jean-Yves Thibon,Nathaniel Thiem,

Vidya Venkateswaran,C.Ryan Vinroot,Ning Yan,Mike Zabrocki

Abstract

We identify two seemingly disparate structures:supercharacters,a useful way of doing

Fourier analysis on the group of unipotent uppertriangular matrices with coecients in a -

nite eld,and the ring of symmetric functions in noncommuting variables.Each is a Hopf

algebra and the two are isomorphic as such.This allows developments in each to be transferred.

The identication suggests a rich class of examples for the emerging eld of combinatorial Hopf

algebras.

1 Introduction

Identifying structures in seemingly disparate elds is a basic task of mathematics.An example,

with parallels to the present work,is the identication of the character theory of the symmetric

group with symmetric function theory.This connection is wonderfully exposited in Macdonald's

book [26].Later,Geissinger and Zelevinsky independently realized that there was an underlying

structure of Hopf algebras that forced and illuminated the identication [19,36].We present a

similar program for a\supercharacter"theory associated to the uppertriangular group and the

symmetric functions in noncommuting variables.

1.1 Uppertriangular matrices

Let UT

n

(q) be the group of uppertriangular matrices with entries in the nite eld F

q

and ones

on the diagonal.This group is a Sylow p-subgroup of GL

n

(q).Describing the conjugacy classes

or characters of UT

n

(q) is a provably\wild"problem.In a series of papers,Andre developed

a cruder theory that lumps together various conjugacy classes into\superclasses"and considers

certain sums of irreducible characters as\supercharacters."The two structures are compatible

(so supercharacters are constant on superclasses).The resulting theory is very nicely behaved |

there is a rich combinatorics describing induction and restriction along with an elegant formula

for the values of supercharacters on superclasses.The combinatorics is described in terms of set

partitions (the symmetric group theory involves integer partitions) and the combinatorics seems

akin to tableau combinatorics.At the same time,supercharacter theory is rich enough to serve as

a substitute for ordinary character theory in some problems [7].

In more detail,the group UT

n

(q) acts on both sides of the algebra of strictly upper-triangular

matrices n

n

(which can be thought of as n

n

= UT

n

(q) 1).The two sided orbits on n

n

can

1

be mapped back to UT

n

(q) by adding the identity matrix.These orbits form the superclasses

in UT

n

(q).A similar construction on the dual space n

n

gives a collection of class functions on

UT

n

(q) that turn out to be constant on superclasses.These orbit sums (suitably normalized) are

the supercharacters.Let

SC =

M

n0

SC

n

;

where SC

n

is the set of functions from UT

n

(q) to C that are constant on superclasses,and SC

0

=

C-spanf1g is by convention the set of class functions of UT

0

(q) = fg.

It is useful to have a combinatorial description of the superclasses in SC

n

.These are indexed by

elements of n

n

with at most one nonzero entry in each row and column.Every superclass contains

a unique such matrix,obtained by a set of elementary row and column operations.Thus,when

n = 3,there are ve such patterns;with 2 F

q

,

0

@

0 0 0

0 0 0

0 0 0

1

A

;

0

@

0 0

0 0 0

0 0 0

1

A

;

0

@

0 0 0

0 0

0 0 0

1

A

;

0

@

0 0

0 0 0

0 0 0

1

A

;and

0

@

0 0

0 0

0 0 0

1

A

:

Each representative matrix X can be encoded as a pair (D;),where D = f(i;j) j X

ij

6= 0g and

:D!F

q

is given by (i;j) = X

ij

.There is a slight abuse of notation here since the pair

(D;) does not record the size of the matrix X.Let X

D;

denote the distinguished representative

corresponding to the pair (D;),and let

D;

=

X

D;

be the function that is 1 on the superclass

and zero elsewhere.

We give combinatorial expressions for the product and coproduct in this section and represen-

tation theoretic descriptions in Section 3.The product is given by

X

D;

X

D

0

;

0

=

X

X

0

X

D;

X

0

0 X

D

0

;

0

;(1.1)

where the sum runs over all ways of placing a matrix X

0

into the upper-right hand block such that

the resulting matrix still has at most one nonzero entry in each row and column.Note that this

diers from the pointwise product of class functions,which is internal to each SC

n

(and hence does

not turn SC into a graded algebra).

For example,if

(D;) = (fg;) $

0 0

0 0

(D

0

;

0

) = (f(1;2);(2;3)g;f(1;2) = a;(2;3) = bg) $

0

@

0 a 0

0 0 b

0 0 0

1

A

;

where the sizes of the matrices are 2 and 3,respectively,then

D;

D

0

;

0 =

0

B

@

0 0 0 0 0

0 0 0 0 0

0 0 0 a 0

0 0 0 0 b

0 0 0 0 0

1

C

A

+

X

c2F

q

0

B

@

0 0 c 0 0

0 0 0 0 0

0 0 0 a 0

0 0 0 0 b

0 0 0 0 0

1

C

A

+

0

B

@

0 0 0 0 0

0 0 c 0 0

0 0 0 a 0

0 0 0 0 b

0 0 0 0 0

1

C

A

:

We can dene the coproduct on SC

n

by

(

X

D;

) =

X

[n]=S[S

c

(i;j)2D only if

i;j 2 S or i;j 2 S

c

(X

D;

)

S

(X

D;

)

S

c

:(1.2)

2

where (X)

S

is the matrix restricted to the rows and columns in S.For example,if D = f(1;4);(2;3)g,

(2;3) = a,and (1;4) = b,then

(

D;

) =

D;

1 +

(

0 a

0 0

)

0 b

0 0

+

0 b

0 0

(

0 a

0 0

)

+1

D;

:

In Section 3,we show that the product and coproduct above have a representation theoretic

meaning and we prove that

Corollary 3.3 With the product (1.1) and the coproduct (1.2),the space SC forms a Hopf algebra.

Background on Hopf algebras is in Section 2.3.We note here that SC is graded,noncommuta-

tive,and cocommutative.It has a unit

;

2 SC

0

and a counit":SC!C obtained by taking the

coecient of

;

.

1.2 Symmetric functions in noncommuting variables

Let be a set partition of [n] = f1;2;:::;ng,denoted `[n].A monomial of shape is a product

of noncommuting variables a

1

a

2

a

k

,where variables are equal if and only if the corresponding

indices/positions are in the same block/part of .For example,if 135j24`[5],then xyxyx is a

monomial of shape (135j24 is the set partition of [5] with parts f1;3;5g and f2;4g).Let m

be

the sum of all monomials of shape .Thus,with three variables

m

135j24

= xyxyx +yxyxy +xzxzx +zxzxz +yzyzy +zyzyz:

Usually,we work with an innite set of variables and formal sums.

Dene

=

M

n0

n

;where

n

= C-spanfm

j `[n]g:

The elements of are called symmetric functions in noncommuting variables.As linear com-

binations of the m

's,they are invariant under permutations of variables.Such functions were

considered by Wolf [34] and Doubilet [16].More recent work of Sagan brought them to the fore-

front.A lucid introduction is given by Rosas and Sagan [30] and combinatorial applications by

Gebhard and Sagan [18].The algebra is actively studied as part of the theory of combinatorial

Hopf algebras [3,9,11,12,22,29].The m

and thus are invariant under permutations of

variables.

Remark.There are a variety of notations given for ,including NCSym and WSym.Instead

of choosing between these two conventions,we will use the more generic ,following Rosas and

Sagan.

Here is a brief denition of product and coproduct;Section 2.4 has more details.If `[k] and

`[n k],then

m

m

=

X

`[n]

^([k]j[nk])=j

m

:(1.3)

where ^ denotes the join in the poset of set partitions under renement (in this poset 1234 precedes

the two incomparable set partitions 1j234 and 123j4),and j `[n] is the set partition

j =

1

j

2

j j

a

j

1

+kj

2

+kj j

b

+k:(1.4)

Thus,if = 1j2 and = 123,then j = 1j2j345.

3

The coproduct is dened by

(m

) =

X

J[`()]

m

st(

J

)

m

st(

J

c

)

;(1.5)

where has`() parts,

J

= f

j

2 j j 2 Jg,st:J![jJj] is the unique order preserving

bijection,and J

c

= [`()] n J.

Thus,

(m

14j2j3

) =m

14j2j3

1 +2m

13j2

m

1

+m

12

m

1j2

+m

1j2

m

12

+2m

1

m

13j2

+1

m

14j2j3

:

It is known ([3,Section 6.2],[11,Theorem 4.1]) that ,endowed with this product and co-

product is a Hopf algebra,where the antipode is inherited from the grading.A basic result of the

present paper is stated here for q = 2 (as described in Section 2.1 below,the pairs (D;) are in

correspondence with set-partitions).The version for general q is stated in Section 3.2.

Theorem 3.2.For q = 2,the function

ch:SC !

7!m

is a Hopf algebra isomorphism.

This construction of a Hopf algebra from the representation theory of a sequence of groups is

the main contribution of this paper.It diers from previous work in that supercharacters are used.

Previous work was conned to ordinary characters (e.g.[25]) and the results of [10] indicate that

this is a restrictive setting.This work opens the possibility for a vast new source of Hopf algebras.

Section 2 gives further background on supercharacters (2.1),some representation theoretic oper-

ations (2.2),Hopf algebras (2.3),and symmetric functions in noncommuting variables (2.4).Section

3 proves the isomorphism theorem for general q,and Section 4 proves an analogous realization for

the dual Hopf algebra.The appendix describes the available Sage programs developed in parallel

with the present study,and a link for a list of open problems.

Acknowledgements

This paper developed during a focused research week at the American Institute of Mathematics in

May 2010.The main results presented here were proved as a group during that meeting.

2 Background

2.1 Supercharacter theory

Supercharacters were rst studied by Andre (e.g.[5]) and Yan [35] in relation to UT

n

(q) in order

to nd a more tractable way to understand the representation theory of UT

n

(q).Diaconis and

Isaacs [15] then generalized the concept to arbitrary nite groups,and we reproduce a version of

this more general denition below.

A supercharacter theory of a nite group G is a pair (K;X) where K is a partition of G and X

is a partition of the irreducible characters of G such that

(a) Each K 2 K is a union of conjugacy classes,

4

(b) f1g 2 K,where 1 is the identity element of G,and f11g 2 X,where 11 is the trivial character

of G.

(c) For X 2 X,the character

X

2X

(1)

is constant on the parts of K,

(d) jKj = jXj.

We will refer to the parts of K as superclasses,and for some xed choice of scalars c

X

2 Q (which

are not uniquely determined),we will refer to the characters

X

= c

X

X

2X

(1) ;for X 2 X

as supercharacters (the scalars c

X

should be picked such that the supercharacters are indeed char-

acters).For more information on the implications of these axioms,including some redundancies in

the denition,see [15].

There are a number of dierent known ways to construct supercharacter theories for groups,

including

Gluing together group elements and irreducible characters using outer automorphisms [15],

Finding normal subgroups N/G and grafting together superchararacter theories for the

normal subgroup N and for the factor group G=N to get a supercharacter theory for the

whole group [21].

This paper will however focus on a technique rst introduced for algebra groups [15],and then

generalized to some other types of groups by Andre and Neto (e.g.[6]).

The group UT

n

(q) has a natural two-sided action on the F

q

-spaces

n = UT

n

(q) 1 and n

= Hom(n;F

q

)

given by left and right multiplication on n and for 2 n

,

(uv)(x 1) = (u

1

(x 1)v

1

);for u;v;x 2 UT

n

(q):

It can be shown that the orbits of these actions parametrize the superclasses and supercharacters,

respectively,for a supercharacter theory.In particular,two elements u;v 2 UT

n

(q) are in the same

superclass if and only if u 1 and v 1 are in the same two-sided orbit in UT

n

(q)nn=UT

n

(q).

Since the action of UT

n

(q) on n can be viewed as applying row and column operations,we obtain

a parameterization of superclasses given by

Superclasses

of UT

n

(q)

!

8

<

:

u 1 2 n with at most

one nonzero entry in

each row and column

9

=

;

:

This indexing set is central to the combinatorics of this paper,so we give several interpretations

for it.Let

M

n

(q) =

(D;)

D f(i;j) j 1 i < j ng;:D!F

q

;

(i;j);(k;l) 2 D implies i 6= k;j 6= l

S

n

(q) =

(

Sets of triples i

a

_j = (i;j;a) 2 [n] [n] F

q

,

with i < j,and i

a

_j;k

b

_l 2 implies i 6= k;j 6= l

)

;

5

where we will refer to the elements of S

n

(q) as F

q

-set partitions.In particular,

M

n

(q) !S

n

(q) !

8

<

:

u 1 2 n with at most

one nonzero entry in

each row and column

9

=

;

(D;) 7! = fi

(i;j)

_ j j (i;j) 2 Dg 7!

X

i

a

_j2

ae

ij

;

(2.1)

where e

ij

is the matrix with 1 in the (i;j) position and zeroes elsewhere.The following table lists

the correspondences for n = 3.

Superclass

0 0 0

0 0 0

0 0 0

!

0 a 0

0 0 0

0 0 0

!

0 0 0

0 0 a

0 0 0

!

0 0 a

0 0 0

0 0 0

!

0 a 0

0 0 b

0 0 0

!

M

3

(q)

D = fg

D = f(1;2)g;

(1;2) = a

D = f(2;3)g;

(2;3) = a

D = f(1;3)g;

(1;3) = a

D = f(1;2);(2;3)g;

(1;2) = a;(2;3) = b

S

3

(q)

1

2

3

1

2

3

a

1

2

3

a

1

2

3

a

1

2

3

a

b

Remark.Consider the maps

:M

n

(q) !M

n

(2)

(D;) 7!(D;1)

and

:S

n

(q)!S

n

(2)

7!fi

1

_j j i

a

_j 2 g:

(2.2)

They ignore the part of the data that involves eld scalars.Note that M

n

(2) and S

n

(2) are in

bijection with the set of partitions of the set f1;2;:::;ng.Indeed,the connected components of an

element 2 S

n

(2) may be viewed as the blocks of a partition of f1;2;:::;ng.Composing the map

with this bijection associates a set partition to an element of M

n

(q) or S

n

(q),which we call the

underlying set partition.

Fix a nontrivial homomorphism#:F

+

q

!C

.For each 2 n

,construct a UT

n

(q)-module

V

= C-spanfv

j 2 UT

n

(q) g

with left action given by

uv

=#

(u

1

1)

v

u

;for u 2 UT

n

(q), 2 UT

n

(q):

It turns out that,up to isomorphism,these modules depend only on the two-sided orbit in

UT

n

(q)nn

=UT

n

(q) of .Furthermore,there is an injective function :S

n

(q)!n

given by

():n !F

q

X 7!

X

i

a

_j2

aX

ij

that maps S

n

(q) onto a natural set of orbit representatives in n

.We will identify 2 S

n

(q) with

() 2 n

.

The traces of the modules V

for 2 S

n

(q) are the supercharacters of UT

n

(q),and they have

a nice supercharacter formula given by

(u

) =

8

>

>

>

<

>

>

>

:

q

#f(i;j;k)ji<j<k;i

a

_k2g

q

#f(i

a

_l;j

b

_k)2ji<j<k<lg

Y

i

a

_l2

i

b

_l2

#(ab);

if i

a

_k 2 and i < j < k

implies i

b

_j;j

b

_k =2 ,

0;otherwise.

(2.3)

6

where u

has superclass type [7].Note that the degree of the supercharacter is

(1) =

Y

i

a

_k2

q

ki1

:(2.4)

Dene

SC =

M

n0

SC

n

;where SC

n

= C-spanf

j 2 S

n

(q)g;

and let SC

0

= C-spanf

;

g.By convention,we write 1 =

;

,since this element will be the identity

of our Hopf algebra.Note that since SC

n

is in fact the space of superclass functions of UT

n

(q),it

also has another distinguished basis,the superclass characteristic functions,

SC

n

= C-spanf

j 2 S

n

(q)g;where

(u) =

1;if u has superclass type ,

0;otherwise,

and

;

=

;

.Section 3 will explore a Hopf structure for this space.

We conclude this section by remarking that with respect to the usual inner product on class

functions

h; i =

1

jUT

n

(q)j

X

u2UT

n

(q)

(u)

(u)

the supercharacters are orthogonal.In fact,for ; 2 S

n

(q),

h

;

i =

q

C()

;where C() =#f(i;j;k;l) j i

a

_k;j

b

_l 2 g:(2.5)

In particular,this inner product remains nondegenerate on SC

n

.

2.2 Representation theoretic functors on SC

We will focus on a number of representation theoretic operations on the space SC.For J =

(J

1

jJ

2

j jJ

`

) any set composition of f1;2;:::;ng,let

UT

J

(q) = fu 2 UT

n

(q) j u

ij

6= 0 with i < j implies i;j are in the same part of Jg:

In the remainder of the paper we will need variants of a straightening map on set compositions.

For each set composition J = (J

1

jJ

2

j jJ

`

),let

st

J

([n]) = st

J

1

(J

1

) st

J

2

(J

2

) st

J

`

(J

`

);(2.6)

where for K [n],st

K

:K ![jKj] is the unique order preserving map.For example,

st

(14j3j256)

([6]) = f1;2g f1g f1;2;3g.

We can extend this straightening map to a canonical isomorphism

st

J

:UT

J

(q) !UT

jJ

1

j

(q) UT

jJ

2

j

(q) UT

jJ

`

j

(q) (2.7)

by reordering the rows and columns according to (2.6).For example,if J = (14j3j256),then

UT

J

(q) 3

0

B

B

B

B

@

1 0 0 a 0 0

0 1 0 0 b c

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 d

0 0 0 0 0 1

1

C

C

C

C

A

st

J

7!

1 a

0 1

;(1);

1 b c

0 1 d

0 0 1

!!

2 UT

2

(q) UT

1

(q) UT

3

(q):

7

Combinatorially,if J = (J

1

jJ

2

j jJ

`

) we let

S

J

(q) = f 2 S

n

(q) j i

a

_j 2 implies i;j are in the same part in Jg:

Then we obtain the bijection

st

J

:S

J

(q) !S

jJ

1

j

(q) S

jJ

2

j

(q) S

jJ

`

j

(q) (2.8)

that relabels the indices using the straightening map (2.6).For example,if J = 14j3j256,then

st

J

1

2

3

4

5

6

a

b

=

1

2

a

1

1

2

3

b

Note that UT

m

(q)UT

n

(q) is an algebra group,so it has a supercharacter theory with the standard

construction [15] such that

SC(UT

m

(q) UT

n

(q))

= SC

m

SC

n

:

The combinatorial map (2.8) preserves supercharacters across this isomorphism.

The rst two operations of interest are restriction

J

Res

UT

n

(q)

st

J

(UT

J

(q))

:SC

n

!SC

jJ

1

j

SC

jJ

2

j

SC

jJ

`

j

7!Res

UT

n

(q)

UT

J

(q)

() st

1

J

;

or

J

Res

UT

n

(q)

st

J

(UT

J

(q))

()(u) = (st

1

J

(u));for u 2 UT

jJ

1

j

(q) UT

jJ

`

j

(q);

and its Frobenius adjoint map superinduction

J

SInd

UT

n

(q)

st

J

(UT

J

(q))

:SC

jJ

1

j

SC

jJ

2

j

SC

jJ

`

j

!SC

n

7!SInd

UT

n

(q)

UT

J

(q)

(st

1

J

());

where for a superclass function of UT

J

(q),

SInd

UT

n

(q)

UT

J

(q)

()(u) =

1

jUT

J

(q)j

2

X

x;y2UT

n

(q)

x(u1)y+12UT

J

(q)

(x(u 1)y +1);for u 2 UT

n

(q):

Note that under the usual inner product on characters,

D

SInd

UT

n

(q)

UT

J

(q)

( );

E

=

D

;Res

UT

n

(q)

UT

J

(q)

()

E

:

Remarks.

(a) While superinduction takes superclass functions to superclass functions,a superinduced char-

acter may not be the trace of a representation.Therefore,SInd is not really a functor on the

module level.An exploration of the relationship between superinduction and induction can

be found in [27].

(b) There is an algorithmic method for computing restrictions of supercharacters (and also tensor

products of characters) [32,33].This has been implemented in Sage (see the Appendix,

below).

8

For an integer composition (m

1

;m

2

;:::;m

`

) of n,let

UT

(m

1

;m

2

;:::;m

`

)

(q) = UT

(1;:::;m

1

jm

1

+1;:::;m

1

+m

2

jjnm

`

+1;:::;n)

(q) UT

m

1

++m

`

(q):

There is a surjective homomorphism :UT

n

(q)!UT

(m

1

;m

2

;:::;m

`

)

(q) such that

2

= ( xes

the subgroup UT

(m

1

;m

2

;:::;m

`

)

(q) and sends the normal complement to 1).The next two operations

arise naturally from this situation.We have in ation

Inf

UT

n

(q)

UT

(m

1

;m

2

;:::;m

`

)

(q)

:SC

m

1

SC

m

2

SC

m

`

!SC

n

;

where

Inf

UT

n

(q)

UT

(m

1

;m

2

;:::;m

`

)

(q)

()(u) = ((u));for u 2 UT

n

(q);

and its Frobenius adjoint map de ation

Def

UT

n

(q)

UT

(m

1

;m

2

;:::;m

`

)

(q)

:SC

n

!SC

m

1

SC

m

2

SC

m

`

;

where

Def

UT

n

(q)

UT

(m

1

;m

2

;:::;m

`

)

(q)

()(u) =

1

j ker()j

X

v2

1

(u)

(v);for u 2 UT

(m

1

;m

2

;:::;m

`

)

(q):

On supercharacters,the in ation map is particularly nice,and is given combinatorially by

Inf

UT

n

(q)

UT

(m

1

;m

2

;:::;m

`

)

(q)

(

1

2

`

) =

1

j

2

jj

`

;

where

1

j

2

j j

`

is as in (1.4) (see for example [32]).

2.3 Hopf algebra basics

Hopf algebras arise naturally in combinatorics and algebra,where there are\things"that break

into parts that can also be put together with some compatibility between operations [23].They

have emerged as a central object of study in algebra through quantum groups [13,17,31] and in

combinatorics [1,4,22].Hopf algebras nd applications in diverse elds such as algebraic topology,

representation theory,and mathematical physics.

We suggest the rst few chapters of [31] for a motivated introduction and [28] as a basic text.

Each has extensive references.The present section gives denitions to make our exposition self-

contained.

Let A be an associative algebra with unit 1 over a eld K.The unit can be associated with a

map

u:K !A

t 7!t 1

A coalgebra is a vector space C over K with two K-linear maps:the coproduct :C!C

C

and a counit":C!K.The coproduct must be coassociative (as a map from C to C

C

C),

so that (

Id) = (Id

) ;or for a 2 C,

X

j

(b

j

)

c

j

=

X

j

b

j

(c

j

);if (a) =

X

j

b

j

c

j

:

9

The counit must be compatible with the coproduct,so that ("

Id) = (Id

") = Id;where

we identify C with K

C and C

K.More explicitly,

a =

X

j

"(b

j

)c

j

=

X

j

b

j

"(c

j

);if (a) =

X

j

b

j

c

j

:

Amap':C!Dbetween coalgebras is a coalgebra map if

D

'= ('

')

C

,where

C

and

D

are the coproducts of C and D,respectively.A subspace I C is a coideal if

C

(I) I

C+C

I

and"(I) = 0.In this case,the quotient space C=I is a coalgebra.

An algebra that is also a coalgebra is a bialgebra if the operations are compatible:for the

coproduct and product,

(xy) = (x)(y) where (a

b)(c

d) = ac

bd;

for the counit and product,

"(xy) ="(x)"(y);

for the unit and coproduct

u = (u

u) ;where

:K !K

K

t 7!t

t;

and for the counit and unit,

" u = Id:

For example,the group algebra K[G] becomes a bialgebra under the maps (g) = g

g and

"(g) = 1 for all g 2 G,and the polynomial algebra K[x

1

;:::;x

n

] becomes a bialgebra under the

operations (x

i

) = x

i

1 +1

x

i

and"(x

i

) = 0 for 1 i n.

A bialgebra is graded if there is a direct sum decomposition

A =

M

n0

A

n

;

such that A

i

A

j

A

i+j

,u(K) A

0

,(A

n

)

L

n

j=0

A

j

A

nj

and"(A

n

) = 0 for all n 1.It is

connected if A

0

= K.For example,the polynomial algebra is graded by polynomial degree.In a

bialgebra,an ideal that is also a coideal is called a biideal,and the quotient is a bialgebra.

A Hopf algebra is a bialgebra with an antipode.This is a linear map S:A!A such that if

(a) =

P

k

a

k

a

0

k

,then

X

k

a

k

S(a

0

k

) ="(a) 1 =

X

k

S(a

k

)a

0

k

:(2.9)

For example,the bialgebra K[G] has antipode S(g) = g

1

and the bialgebra K[x

1

;:::;x

n

] has

antipode S(x

i

) = x

i

.More generally,if A is a connected,graded bialgebra,then (2.9) can be

solved inductively to give S(t 1) = t 1 for t 1 2 A

0

,and for a 2 A

n

,

S(a) = a

n1

X

j=1

S(a

j

)a

0

nj

;where (a) = a

1 +1

a +

n1

X

j=1

a

j

a

0

nj

:(2.10)

Thus,any graded,connected bialgebra has an antipode and is automatically a Hopf algebra.

If A is a graded bialgebra (Hopf algebra) and each A

n

is nite-dimensional,then the graded

dual

A =

M

n0

A

n

is also a bialgebra (Hopf algebra).If Ais commutative (cocommutative),then A

is cocommutative

(commutative).

10

2.4 The Hopf algebra

Symmetric polynomials in a set of commuting variables X are the invariants of the action of the

symmetric group S

X

of X by automorphisms of the polynomial algebra K[X] over a eld K.

When X = fx

1

;x

2

;:::g is innite,we let S

X

be the set of bijections on X with nitely many

nonxed points.Then the subspace of K[[X]]

S

X

of formal power series with bounded degree is the

algebra of symmetric functions Sym(X) over K.It has a natural bialgebra structure dened by

(f) =

X

k

f

0

k

f

00

k

;(2.11)

where the f

0

k

;f

00

k

are dened by the identity

f(X

0

+X

00

) =

X

k

f

0

k

(X

0

)f

00

k

(X

00

);(2.12)

and X

0

+X

00

denotes the disjoint union of two copies of X.The advantage of dening the coproduct

in this way is that is clearly coassociative and that it is obviously a morphism for the product.

For each integer partition = (

1

;

2

;:::;

`

),the monomial symmetric function corresponding to

is the sum

m

(X) =

X

x

2O(x

)

x

(2.13)

over elements of the orbit O(x

) of x

= x

1

1

x

2

2

x

`

`

under S

X

,and the monomial symmetric

functions form a basis of Sym(X).The coproduct of a monomial function is

(m

) =

X

[=

m

m

:(2.14)

The dual basis m

of m

is a multiplicative basis of the graded dual Sym

,which turns out to be

isomorphic to Sym via the identication m

n

= h

n

(the complete homogeneous function,the sum

of all monomials of degree n).

The case of noncommuting variables is very similar.Let A be an alphabet,and consider the

invariants of S

A

acting by automorphisms on the free algebra KhAi.Two words a = a

1

a

2

a

n

and b = b

1

b

2

b

n

are in the same orbit whenever a

i

= a

j

if and only if b

i

= b

j

.Thus,orbits

are parametrized by set partitions in at most jAj blocks.Assuming as above that A is innite,we

obtain an algebra based on all set partitions,dening the monomial basis by

m

(A) =

X

w2O

w;(2.15)

where O

is the set of words such that w

i

= w

j

if and only if i and j are in the same block of .

One can introduce a bialgebra structure by means of the coproduct

(f) =

X

k

f

0

k

f

00

k

where f(A

0

+A

00

) =

X

k

f

0

k

(A

0

)f

00

k

(A

00

);(2.16)

and A

0

+A

00

denotes the disjoint union of two mutually commuting copies of A.The coproduct of

a monomial function is

(m

) =

X

J[`()]

m

st(

J

)

m

st(

J

c)

:(2.17)

This coproduct is cocommutative.With the unit that sends 1 to m

;

and the counit"(f(A)) =

f(0;0;:::),we have that is a connected graded bialgebra and therefore a graded Hopf algebra.

Remark.We again note that is often denoted in the literature as NCSym or WSym.

11

3 A Hopf algebra realization of SC

This section explicitly denes the Hopf structure on SC from a representation theoretic point of

view.We then work out the combinatorial consequences of these rules,and it directly follows

that SC

=

for q = 2.We then proceed to yield a\colored"version of that will give the

corresponding Hopf structure for the other values of q.

3.1 The correspondence between SC and

In this section we describe a Hopf structure for the space

SC =

M

n0

SC

n

= C-spanf

j 2 S

n

(q);n 2 Z

0

g

= C-spanf

j 2 S

n

(q);n 2 Z

0

g:

The product and coproduct are dened representation theoretically by the in ation and restriction

operations of Section 2.2,

= Inf

UT

a+b

(q)

UT

(a;b)

(q)

( );where 2 SC

a

; 2 SC

b

;(3.1)

and

() =

X

J=(AjA

c

)

A[n]

J

Res

UTn(q)

UT

jAj

(q)UT

jA

c

j

(q)

();for 2 SC

n

:(3.2)

For a combinatorial description of the Hopf structure of SC it is most convenient to work with the

superclass characteristic functions.A matrix description appears in (1.1) and (1.2).

Proposition 3.1.

(a) For 2 S

k

(q), 2 S

nk

(q),

=

X

=t t(k+)2S

n

(q)

i

a

_l2 implies ik<l

;

where (k +) = f(k +i)

a

_(k +j) j i

a

_j 2 g and t denotes disjoint union.

(b) For 2 S

n

(q),

(

) =

X

=t

2S

A

(q);2S

A

c

(q)

Af1;2;:::;ng

st

A

()

st

A

c()

:

Proof.(a) Let (u

1) 2 n

n

be the natural orbit representative for the superclass corresponding

to .Then

Inf

UT

n

(q)

UT

k

(q)UT

nk

(q)

(

)(u

) = (

)((u

));

where

:UT

n

(q) !UT

k

(q) UT

nk

(q)

A

C

0

B

7!

A

0

0

B

:

12

Thus,

Inf

UT

n

(q)

UT

k

(q)UT

nk

(q)

(

)(u

) =

8

<

:

1;

if = fi

a

_j 2 j i;j 2 [k]g,

and +k = fi

a

_j 2 j i;j 2 [k]

c

g,

0;otherwise,

as desired.

(b) Let 2 S

A

(q),and 2 S

A

c

(q),and let u

be the corresponding superclass representative

for UT

AjA

c(q).Note that

Res

UT

n

(q)

UT

AjA

c(q)

(

)(u

) =

1;if

(u

) = 1,

0;otherwise,

=

8

<

:

1;

if = fi

a

_j 2 j i;j 2 Ag and

= fi

a

_j 2 j i;j 2 A

c

g;

0;otherwise.

Thus,if = fi

a

_j 2 j i;j 2 Ag and = fi

a

_j 2 j i;j 2 A

c

g,then

Res

UT

n

(q)

UT

AjA

c

(q)

(

) =

;

and the result follows by applying the st

J

map.

Example.We have

1

2

3

a

1

2

3

4

b

c

=

1

2

3

4

5

6

7

a

b

c

+

X

d2F

q

1

2

3

4

5

6

7

a

d

b

c

+

1

2

3

4

5

6

7

a

d

b

c

+

1

2

3

4

5

6

7

a

d

b

c

+

1

2

3

4

5

6

7

a

d

b

c

+

X

d;e2F

q

1

2

3

4

5

6

7

a

d

e

b

c

+

1

2

3

4

5

6

7

a

d

e

b

c

:

and

1

2

3

4

a

=

1

2

3

4

a

;

+2

1

2

3

a

1

+

1

2

a

1

2

+

1

2

1

2

a

+2

1

1

2

3

a

+

;

1

2

3

4

a

:

By comparing Proposition 3.1 to (1.3) and (2.17),we obtain the following theorem.

Theorem 3.2.For q = 2,the map

ch:SC !

7!m

is a Hopf algebra isomorphism.

Note that although we did not assume for the theorem that SC is a Hopf algebra,the fact that

ch preserves the Hopf operations implies that SC for q = 2 is indeed a Hopf algebra.The general

result will follow from Section 3.2.

13

Corollary 3.3.The algebra SC with product given by (3.1) and coproduct given by (3.2) is a Hopf

algebra.

Remarks.

(a) Note that the isomorphism of Theorem 3.2 is not in any way canonical.In fact,the automor-

phism group of is rather large,so there are many possible isomorphisms.For our chosen

isomorphism,there is no nice interpretation for the image of the supercharacters under the

isomorphism of Theorem 3.2.Even less pleasant,when one composes it with the map

!Sym

that allows variables to commute (see [16,30]),one in fact obtains that the supercharacters

are not Schur positive.But,exploration with Sage suggests that it may be possible to choose

an isomorphism such that the image of the supercharacters are Schur positive.

(b) Although the antipode is determined by the bialgebra structure of ,explicit expressions

are not well understood.However,there are a number of forthcoming papers (e.g.[2,24])

addressing this situation.

(c) In [1],the authors considered the category of combinatorial Hopf algebras consisting of pairs

(H;),where H is a graded connected Hopf algebra and :H!C is a character (an algebra

homomorphism).As remarked in [10],every graded Hopf algebra arising from representation

theory yields a canonical character.This is still true for SC.For all n 0 consider the dual

to the trivial supercharacter (

;

n

)

2 SC

n

.It follows from Section 4.1 below that

(

;

n

)

=

n

X

k=0

(

;

k

)

(

;

nk

)

;

which implies that the map :SC!C given by

() = h(

;

n

)

;i;where 2 SC

n

,

is a character.We thus have that (SC;) is a combinatorial Hopf algebra in the sense of [1].

This connection awaits further exploration.

The Hopf algebra SChas a number of natural Hopf subalgebras.One of particular interest is the

subspace spanned by linear characters (characters with degree 1).In fact,for this supercharacter

theory every linear character of U

n

is a supercharacter and by (2.4) these are exactly indexed by

the set

L

n

= f 2 S

n

(q) j i

a

_j 2 implies j = i +1g:

Corollary 3.4.For q = 2,the Hopf subalgebra

LSC = C-spanf

j i

1

_j 2 implies j = i +1g;

is isomorphic to the Hopf algebra of noncommutative symmetric functions Sym studied in [20].

Proof.Let the length of an arc i

a

_j be j i.By inspection of the product and coproduct of SC,we

observe that an arc i

a

_j never increases in length.Since LSC is the linear span of supercharacters

indexed by set partitions with arcs of length at most 1,it is clearly a Hopf subalgebra.

14

By (2.3),for [n] = fi

1

_(i +1) j 1 i < ng,we have h

;

[n]

i = 0 unless 2 L

n

.Thus,the

superclass functions

[n]

2 LSC.Furthermore,if we order by renement in SC

n

,then the set of

products

f

[k

1

]

[k

2

]

[k

`

]

j k

1

+k

2

+ +k

`

= n;` 1g

have an upper-triangular decomoposition in terms of the

.Therefore,the elements

[n]

are

algebraically independent in LSC,and LSC contains the free algebra

Ch

[1]

;

[2]

;:::i:

Note that every element 2 L

n

is of the form

= [1

_

_k

1

] [

(k

1

+1)

_

_(k

1

+k

2

)

[:::[

(n k

`

)

_

_n

;

where [i

_

_j] = fi

1

_(i +1);(i +1)

1

_(i +2);:::;(j 1)

1

_jg:Thus,

jL

n

j = dim

C-spanf

[k

1

]

[k

2

]

[k

`

]

j k

1

+k

2

+ +k

`

= n;` 1g

;

implies

LSC = Ch

[1]

;

[2]

;:::i:

On the other hand,[20] describes Sym as follows:

Sym= Ch

1

;

2

;:::i

is the free (non-commutative) algebra with deg(

k

) = k and coproduct given by

(

k

) = 1

k

+

k

1:

Hence,the map

[k]

7!

k

gives the desired isomorphism.

Remark.In fact,for each k 2 Z

0

the space

SC

(k)

= C-spanf

j i_j 2 implies j i kg

is a Hopf subalgebra.This gives an unexplored ltration of Hopf algebras which interpolate between

LSC and SC.

3.2 A colored version of

There are several natural ways to color a combinatorial Hopf algebra;for example see [8].The

Hopf algebra SC for general q is a Hopf subalgebra of the\naive"coloring of .

Let C

r

= hi be a cyclic group of order r (which in our case will eventually be r = q 1).We

expand our set of variables A = fa

1

;a

2

;:::g by letting

A

(r)

= AC

r

:

We view the elements of C

r

as colors that decorate the variables of A.The group S

A

acts on the

rst coordinate of the set A

(r)

.That is,(a

i

;

j

) = ((a

i

);

j

).With this action,we dene

~

(r)

as the set of bounded formal power series in A

r

invariant under the action of S

A

.As before,we

assume that A is innite and the space

~

(r)

is a graded algebra based on r-colored set partitions

15

(;(

1

;:::;

n

)) where is a set partition of the set f1;2;:::;ng and (

1

;:::;

n

) 2 C

n

r

.It has a

basis of monomial elements given by

m

;(

1

;:::;

n

)

A

(r)

=

X

w2O

;(

1

;:::;

n

)

w;(3.3)

where O

;(

1

;:::;

n

)

is the orbit of S

A

indexed by (;(

1

;:::;

n

)).More precisely,it is the set of

words w = (a

i

1

;

1

)(a

i

2

;

2

):::(a

i

n

;

n

) on the alphabet A

(r)

such that a

i

= a

j

if and only if i

and j are in the same block of .The concatenation product on KhA

(r)

i gives us the following

combinatorial description of the product in

~

(r)

in the monomial basis.If `[k] and `[n k],

then

m

;(

1

;:::;

k

)

m

;(

0

1

;:::;

0

nk

)

=

X

`[n]

^([k]j[nk])=j

m

;(

1

;:::;

k

;

0

1

;:::;

0

nk

)

:(3.4)

This is just a colored version of (1.3).

As before,we dene a coproduct by

(f) =

X

k

f

0

k

f

00

k

where f

A

0(r)

+A

00(r)

=

X

k

f

0

k

A

0(r)

f

00

k

A

00(r)

(3.5)

and A

0(r)

+A

00(r)

denotes the disjoint union of two mutually commuting copies of A

(r)

.This is clearly

coassociative and a morphism of algebras;hence,

r

is a graded Hopf algebra.The coproduct of

a monomial function is

m

;(

1

;:::;

n

)

=

X

_=

m

st();

m

st();

;(3.6)

where

denotes the subsequence (

i

1

;

i

2

;:::) with i

1

< i

2

< and i

j

appearing in a block of .

The complement sequence is

.This coproduct is cocommutative.With the unit u:1 7!1 and

the counit :f(A

(r)

) 7!f(0;0;:::) we have that

~

(r)

is a connected graded bialgebra and therefore

a graded Hopf algebra.

Now we describe a Hopf subalgebra of this space indexed by S

n

(q) for n 0.For (D;) 2 S

n

(q),

let

k

(D;)

=

X

(

1

;:::;

n

)2C

n

r

j

=

i

=(i;j)

m

(D;);(

1

;:::;

n

)

;

where (D;) is the underlying set partition of D (as in (2.2)).

Proposition 3.5.The space

(q1)

= C-spanfk

(D;)

j (D;) 2 M

n

(q);n 2 Z

0

g

is a Hopf subalgebra of

~

(q1)

.For 2 S

k

(q), 2 S

nk

(q) the product is given by

k

k

=

X

=t t(k+)2S

n

(q)

i

a

_l2 implies ik<l

k

;(3.7)

and for 2 S

n

(q),the coproduct is given by

(k

) =

X

=t

2S

A

(q);2S

A

c

(q)

Af1;2;:::;ng

k

st

A

()

k

st

A

c()

:(3.8)

16

Proof.It is sucient to show that

(q1)

is closed under product and coproduct.Thus,it is enough

to show that (3.7) and (3.8) are valid.For = (D;) 2 S

k

(q), = (D

0

;

0

) 2 S

nk

(q) let () and

() be the underlying set partitions of and ,respectively.We have

k

k

=

X

(

1

;:::;

k

)2C

k

r

j

=

i

=(i;j)

m

();(

1

;:::;

k

)

X

(

k+1

;:::;

n

)2C

nk

r

k+j

=

k+i

=

0

(i;j)

m

();(

k+1

;:::;

n

)

=

X

(

1

;:::;

n

)2C

n

r

j

=

i

=(i;j);i<jk

j

=

i

=

0

(ik;jk);k<i<j

X

`[n]

^([k]j[nk])=()j()

m

;(

1

;:::;

n

)

=

X

=t t(k+)2S

n

(q)

i

a

_l2 implies ik<l

k

:

In the second equality,the second sum ranges over set partitions obtained by grouping some

block of () with some block of ().These set partitions can be thought of as collections of arcs

i _j with 1 i k < j n.In the last equality,we group together the terms m

;(

1

;:::;

n

)

such

that

j

=

i

=

00

(i;j) for i k < j.

Now for = (D;) 2 S

n

(q),

(k

) =

X

(

1

;:::;

n

)2C

n

r

j

=

i

=(i;j)

m

();(

1

;:::;

n

)

=

X

(

1

;:::;

n

)2C

n

r

j

=

i

=(i;j)

X

_=

m

st();

m

st();

=

X

=t

2S

A

(q);2S

A

c

(q)

Af1;2;:::;ng

k

st

A

()

k

st

A

c()

:

Comparing Proposition 3.5 and Proposition 3.1,we obtain

Theorem 3.6.The map

ch:SC !

(q1)

7!k

(D

;

)

is an isomorphism of Hopf algebras.In particular,SC is a Hopf algebra for any q.

Remark.As in the q = 2 case,for each k 2 Z

0

the space

SC

(k)

= C-spanf

j i

a

_j 2 implies j i kg

is a Hopf subalgebra of SC.For k = 1,this gives a q-version of the Hopf algebra of noncommutative

symmetric functions..

4 The dual Hopf algebras SC

and

This section explores the dual Hopf algebras SC

and

.We begin by providing representation

theoretic interpretations of the product and coproduct of SC

,followed by a concrete realization

of SC

and

.

17

4.1 The Hopf algebra SC

As a graded vector space,SC

is the vector space dual to SC:

SC

=

M

n0

SC

n

= C-spanf

j 2 S

n

(q);n 2 Z

0

g

= C-spanf(

)

j 2 S

n

(q);n 2 Z

0

g:

We may use the inner product (2.5) to identify SC

with SC as graded vector spaces.Under this

identication,the basis element dual to

with respect to the inner product (2.5) is

= z

;where z

=

jUT

jj

(q)j

jUT

jj

(q)(u

1)UT

jj

(q)j

;

and the basis element dual to

is

(

)

= q

C()

;where C() =#f(i;j;k;l) j i

a

_k;j

b

_l 2 g:

The product on SC

is given by

=

X

J=AjA

c

A[a+b]

jAj=a

J

SInd

UT

a+b

(q)

UT

a

(q)UT

b

(q)

( );for 2 SC

a

; 2 SC

b

;

and the coproduct by

() =

n

X

k=0

Def

UT

n

(q)

UT

(k;nk)

(q)

();for 2 SC

n

:

Proposition 4.1.The product and coproduct of SC

in the

basis is given by

(a) for 2 S

k

(q) and 2 S

nk

(q),

=

X

J[n]

jJj=k

st

1

J

()[st

1

J

c

()

;

(b) for 2 S

n

(q),

(

) =

n

X

k=0

[k]

[k]

c

;where

J

= fi

a

_j 2 j i;j 2 Jg:

Proof.This result follows from Proposition 3.1,and the duality results

D

J

SInd

UT

n

(q)

st

J

(UT

J

(q))

( );

E

=

D

;

J

Res

UT

n

(q)

st

J

(UT

J

(q))

()

E

,

D

Inf

UT

n

(q)

UT

(m

1

;:::;m

`

)

(q)

( );

E

=

D

;Def

UT

n

(q)

UT

(m

1

;:::;m

`

)

(q)

()

E

,

h

;

i =

.

Note that the Hopf algebra SC

is commutative,but not cocommutative.

18

Example.We have

1

2

a

1

2

3

b

=

1

2

3

4

5

a

b

+

1

2

3

4

5

a

b

+

1

2

3

4

5

a

b

+

1

2

3

4

5

a

b

+

1

2

3

4

5

b

a

+

1

2

3

4

5

b

a

+

1

2

3

4

5

b

a

+

1

2

3

4

5

b

a

+

1

2

3

4

5

b

a

+

1

2

3

4

5

b

a

and

1

2

3

4

a

b

=

1

2

3

4

a

b

;

+

1

2

3

a

1

+

1

2

a

1

2

+

1

1

2

3

b

+

;

1

2

3

4

a

b

:

4.2 A realization of SC

Apriori,it is not clear that

or SC

should have a realization as a space of functions in commuting

variables.Here,we summarize some results of [22] giving such a realization,and remark that the

variables must satisfy relations closely related to the denition of S

n

(q).

Let x

ij

,for i;j 1,be commuting variables satisfying the relations

x

ij

x

ik

= 0 and x

ik

x

jk

= 0 for all i;j;k.(4.1)

For a permutation 2 S

n

,dene

M

=

X

i

1

<<i

n

x

i

1

i

(1)

x

i

n

i

(n)

:(4.2)

It is shown in [22] that these polynomials span a (commutative,cofree) Hopf algebra,denoted by

SQSym.

For 2 S

m

and 2 S

n

we dene coecients C

;

M

M

=

X

C

;

M

:(4.3)

which can be computed by the following process:

Step 1.Write and as products of disjoint cycles.

Step 2.For each subset A [m + n] with m elements,renumber using the unique order-

preserving bijection st

1

A

:[m]!A,and renumber with the unique order preserving

bijection st

1

A

c

:[n]!A

c

.

Step 3.The resulting permutation gives a term M

in the product M

M

.

Thus,C

;

is the number of ways to obtain from and using this process.

Example.If = (1)(2) = 12, = (31)(2) = 321,then Step 2 yields

A=f1;2g

(1)(2)(53)(4);

A=f1;3g

(1)(3)(52)(4);

A=f1;4g

(1)(4)(52)(3);

A=f1;5g

(1)(5)(42)(3);

A=f2;3g

(2)(3)(51)(4);

A=f2;4g

(2)(4)(51)(3);

A=f2;5g

(2)(5)(41)(3);

A=f3;4g

(3)(4)(51)(2);

A=f3;5g

(3)(5)(41)(2);

A=f4;5g

(4)(5)(31)(2);

(4.4)

and thus C

(51)(2)(3)(4)

(1)(2);(31)(2)

= 3.

19

Another interpretation of this product is given by the dual point of view:C

;

is the number

of ways of getting (;) as the standardized words of pairs (a;b) of two complementary subsets of

cycles of .For example,with = 12, = 321,and = 52341,one has three solutions for the

pair (a;b),namely

((2)(3);(4)(51));((2)(4);(3)(51));((3)(4);(2)(51)):(4.5)

Remark.Each function in SQSymcan be interpreted as a function on matrices by evaluating x

ij

at the (i;j)-th entry of the matrix (or zero if the matrix does not have an (i;j) entry).From this

point of view,SQSym intersects the ring of class functions of the wreath product C

r

o S

n

in such

a way that it contains the ring of symmetric functions as a natural subalgebra.

For a permutation 2 S

n

,let csupp() be the partition of the set [n] whose blocks are the

supports of the cycles of .The sums

U

:=

X

csupp()=

M

(4.6)

span a Hopf subalgebra QSym of SQSym,which is isomorphic to the graded dual of .Indeed,

from the product rule of the M

given in (4.3),it follows that

U

U

:=

X

C

;

U

;(4.7)

where C

;

is the number of ways of splitting the parts of into two subpartitions whose standard-

ized words are and .For example,

U

124j3

U

1

= U

124j3j5

+2U

125j3j4

+U

135j4j2

+U

235j4j1

:(4.8)

Hence,the basis U

has the same product rule as m

.However,it does not have the same

coproduct.To nd the correct identication we need the p

basis introduced in [30].Let

p

=

X

m

;

where means that renes .As shown in [9,30]

p

p

= p

j

and following the notation of (1.5)

(p

) =

X

J[`()]

p

st(

J

)

p

st(

J

c)

:

These are precisely the operations we need to give the isomorphism

:QSym !

U

7!p

:

The isomorphism in Theorem 3.2 maps

7!m

,and the dual map is m

7!

.Hence,if we

dene

V

=

1

(m

) =

X

U

;

then we obtain the following theorem.

20

Theorem 4.2.For q = 2,the function

ch:SC

!QSym

7!V

is a Hopf algebra isomorphism.

Remark.For general q one needs a colored version of QSym.This can be done in the same

spirit of Section 3.2 and we leave it to the reader.

5 Appendix

In addition to the above results,the American Institute of Mathematics workshop generated several

items that might be of interest to those who would like to pursue these thoughts further.

5.1 Sage

A Sage package has been written,and is described at

http://garsia.math.yorku.ca/~saliola/supercharacters/

It has a variety of functions,including the following.

It can use various bases,including the supercharacter basis and the superclass functions basis,

It can change bases,

It computes products,coproducts and antipodes in this Hopf algebra,

It computes the inner tensor products (pointwise product) and restriction in the ring of

supercharacters,

It gives the supercharacter tables for UT

n

(q).

5.2 Open problems

There is a list of open problems related to this subject available at

http://www.aimath.org/pastworkshops/supercharacters.html

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M.Aguiar,University of Texas A&M,maguiar@math.tamu.edu

C.Andre,University of Lisbon,caandre@fc.ul.pt

C.Benedetti,York University,carobene@mathstat.yorku.ca

N.Bergeron,York University,supported by CRC and NSERC,bergeron@yorku.ca

Z.Chen,York University,czhi@mathstat.yorku.ca

P.Diaconis,Stanford University,supported by NSF DMS-0804324,diaconis@math.stanford.edu

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S.Hsiao,Bard College,hsiao@bard.edu

I.M.Isaacs,University of Wisconsin-Madison,isaacs@math.wisc.edu

A.Jedwab,University of Southern California,supported by NSF DMS 07-01291,jedwab@usc.edu

K.Johnson,Penn State Abington,kwj1@psuvm.psu.edu

G.Karaali,Pomona College,gizem.karaali@pomona.edu

A.Lauve,Loyola University,lauve@math.luc.edu

T.Le,University of Aberdeen,t.le@abdn.ac.uk

S.Lewis,University of Washington,supported by NSF DMS-0854893,stedalew@u.washington.edu

H.Li,Drexel University,supported by NSF DMS-0652641,huilan.li@gmail.com

K.Magaard,University of Birmingham,k.magaard@bham.ac.uk

E.Marberg,MIT,supported by NDSEG Fellowship,emarberg@math.mit.edu

J.-C.Novelli,Universite Paris-Est Marne-la-Vallee,novelli@univ-mlv.fr

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L.Tevlin,New York University,ltevlin@nyu.edu

J.-Y.Thibon,Universite Paris-Est Marne-la-Vallee,jyt@univ-mlv.fr

N.Thiem,University of Colorado at Boulder,supported by NSF DMS-0854893,thiemn@colorado.edu

V.Venkateswaran,Caltech University,vidyav@caltech.edu

C.R.Vinroot,College of William and Mary,supported by NSF DMS-0854849,vinroot@math.wm.edu

N.Yan,ning.now@gmail.com

M.Zabrocki,York University,zabrocki@mathstat.yorku.ca

23

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