DSpace at VNU: SUPERCHARACTERS AND PATTERN SUBGROUPS IN THE UPPER TRIANGULAR GROUPS

SUPERCHARACTERS AND PATTERN SUBGROUPS IN THE UPPER TRIANGULAR GROUPS TUNG LE∗ Faculty of Mathematics and Computer Science, Vietnam National University, Ho Chi Minh City, Vietnam lttung96

SUPERCHARACTERS AND PATTERN SUBGROUPS IN

THE UPPER TRIANGULAR GROUPS

TUNG LE∗ Faculty of Mathematics and Computer Science, Vietnam National University,

Ho Chi Minh City, Vietnam (lttung96@yahoo.com)

Abstract Let U n (q) denote the upper triangular group of degree n over the finite field Fq with q

elements It is known that irreducible constituents of supercharacters partition the set of all irreducible

characters Irr(U n (q)) In this paper we present a correspondence between supercharacters and pattern subgroups of the form U k (q) ∩ w U k (q), where w is a monomial matrix in GL k (q) for some k < n Keywords: root system; irreducible character; triangular group

2010 Mathematics subject classification: Primary 20C33; 20C15

1 Introduction

Let q be a power of a prime p and let Fq be a field with q elements The group U n (q)

of all upper triangular (n × n)-matrices over F q with all diagonal entries equal to 1 is

a Sylow p-subgroup of GL n(Fq) It was conjectured by Higman [8] that the number

of conjugacy classes of U n (q) is given by a polynomial in q with integer coefficients.

Isaacs [10] showed that the degrees of all irreducible characters of U n (q) are powers of

q Huppert [9] proved that character degrees of U n (q) are precisely of the form {q e: 0

e
µ(n)}, where the upper bound µ(n) was known to Lehrer [13] Lehrer conjectured

that each number N n,e (q) of irreducible characters of U n (q) of degree q e is given by

a polynomial in q with integer coefficients Isaacs [11] suggested a strengthened form

of Lehrer’s Conjecture, stating that N n,e (q) is given by a polynomial in (q − 1) with

non-negative integer coefficients So, Isaacs’s Conjecture implies Higman’s and Lehrer’s Conjectures

Many efforts have been made to understand more about U n (q); see [1, 3, 5, 7, 10,

11, 14, 15], among others Supercharacters arise as tensor products of some elementary

characters to give a ‘nice’ partition of all non-principal irreducible characters of U n (q)

(see [1, 12]) Supercharacters have been defined for Sylow p-subgroups of other finite

groups of Lie type (see [2]), and in general for algebra groups (see [5]).

∗ Present address: School of Mathematical Science, North-West University, Mafikeng Campus, South

Africa.

c

2012 The Edinburgh Mathematical Society 177

Here, for U n (q) we show a natural correspondence between supercharacters and pattern

subgroups (Theorem 2.8) To highlight the main idea of construction, we have deferred all of our proofs to§ 3.

2 Supercharacters and pattern subgroups

Let Σ = Σ n −1 =
α1, , α n −1  be the root system of GL n (q) with respect to the

maximal split torus equal to the diagonal group (see [4, Chapter 3]) Set α i,j = α i+

α i+1+· · · + α j for all 0 < i
j < n Denote by Σ+ the set of all positive roots The

root subgroup X α i,j is the set of all matrices of the form I n + c · e i,j+1 , where I n = the

identity (n × n)-matrix, c ∈ F q and e i,j+1 is equal to the zero matrix except for a ‘1’ at

entry (i, j + 1) The upper triangular group U n (q) is generated by all X α , where α ∈ Σ+

We write U for U n (q) if n and q are clear from the context For convenience when using

the root system, we consider the upper triangular group as a tableaux:

⎛

⎜

⎜

⎜

⎝

· 1 ∗ ∗ ∗

· · 1 ∗ ∗

· · · 1 ∗

· · · · 1

⎞

⎟

⎟

⎟→

α1 α 1,2 α 1,3 α 1,4

α2 α 2,3 α 2,4

α3 α 3,4

α4

A subset S ⊂ Σ+is called closed if, for each α, β ∈ S such that α+β ∈ Σ+, α + β ∈ S.

A pattern subgroup of U is a group generated by all root subgroups X α , where α ∈ S a

closed positive root subset

Let G be a group Set G × = G \{1} Denote by Irr(G) the set of all complex irreducible characters of G, and let Irr(G) × = Irr(G) \{1 G } For H  G, let Irr(G/H) denote the set

of all irreducible characters of G with H in the kernel If K
G such that G = H
K, then for each character ξ of K we denote the inflation of ξ to G by ξ G , i.e ξ G is the

extension of ξ to G with H ⊂ ker(ξ G ) Furthermore, for H
G and ξ ∈ Irr(H), we define

by Irr(G, ξ) = {χ ∈ Irr(G): (χ, ξ G)= 0} the irreducible constituent set of ξ G, and for

χ ∈ Irr(G) we denote its restriction to H by χ| H

For a field K, let K ×be its multiplicative group In the whole paper, we fix a non-trivial

linear character ϕ : (Fq , +) → C × For each α ∈ Σ+and s ∈ F q , the map φ α,s : X α → C ×,

x α (d) → ϕ(ds) is a linear character of the root subgroup X α, and all linear characters of

X α arise in this way

For each α i,j, we define

arm(α i,j) ={α i,k : i
k < j} and leg(α i,j) ={α k,j : i < k
j}.

If i = j, α i,i = α i , then arm(α i ) and leg(α i ) are empty For each α ∈ Σ+, we define the

hook of α as h(α) = arm(α) ∪ leg(α) ∪ {α}, the hook group of α as H α=
X β : β ∈ h(α), and the base group V α =
X β : β ∈ Σ+\ arm(α) Since [V α , V α]∩ X α ={1}, for each

s ∈ F ×

q there exists a linear λ α,s ∈ Irr(V α ) such that λ α,s | X α = φ α,s and λ α,s | X β = 1X β

for other root subgroups X β ⊂ V α , β = α Denote by Irr(V α /[V α , V α])× the set of all

these linear characters of V

Lemma 2.1 λ U α,s is irreducible for all s ∈ F ×

q

We call λ U α,s an elementary character of U associated to α A basic set D is a non-empty subset of Σ+ in which none of the roots are in the same row or column For each

basic set D, define

E(D) = 

α ∈D

Irr(V α /[V α , V α])× .

For each basic set D and φ ∈ E(D), we define a supercharacter, also known as basic

character in [1],

ξ D,φ= 

λ α,s ∈φ

λ U α,s

It turns out that each supercharacter ξ D,φis induced from a linear character of a pattern subgroup

Definition 2.2 We define

V D=

α ∈D

V α and λ D= 

λ α,s ∈φ

λ α,s | V D

Lemma 2.3 We have ξ D,φ = λ U

D

It is easy to see that V D is generated by all X β , where β ∈ Σ+\ ( α ∈D arm(α)), and

λ D is a linear character of V D For each basic set D, it can be proven that the diagonal

subgroup of GLn (q) acts transitively on E(D) by conjugation So it makes sense when

we write λ D here instead of λ D,φ , and it also says that the decomposition of ξ D,φ is

dependent only on D To know more about supercharacters, see, for example, [5, 6].

Here, we recall the main role of supercharacters as a partition of Irr(U ) ×.

Theorem 2.4 For each χ ∈ Irr(U) × , there exist uniquely a basic set D and φ ∈ E(D) such that χ is an irreducible constituent of ξ D,φ

Denote by Irr(ξ D,φ ) the set of all irreducible constituents of ξ D,φ Here, to prove Hig-man’s Conjecture, it suffices to prove that|Irr(ξ D,φ)| is a polynomial in q.

Now for each basic set D of size k = |D|, we define an associated monomial (k × k)-matrix w D ∈ GL k (q) First of all, we define two partial orders on Σ+

Definition 2.5 We define <rand <bon Σ+ as follows:

(i) α i,j <rα l,k if j < k (i.e to the right);

(ii) α < α if i < l (i.e to the bottom).

i, j i, j i, j

Figure 1 Positions of ν i,j and γ i,j

An easy way to understand these two orders is <rstanding for left to right and <bfor

top to bottom It is noted that, on a basic set, <r and <b are total orders

Now we fix a basic set D of size k ascending order of <r Let D = {τ1, , τ k }, where

τ i <r τ j if i < j We define w D = (a i,j)∈ GL k (q) as follows:

a i,j=

1 if τ j is the ith element of D in ascending order <b,

0 otherwise.

For example, if D = {α2,3, α1,4, α3,5 }, |D| = 3,

α1,4 α2,3

α3,5

then

w D=

⎛

⎜

⎝

0 1 0

1 0 0

0 0 1

⎞

⎟

⎠

It is clear that w D is a monomial matrix in the Weyl group S k of GLk (q) Here, w D

somehow gives pivots of D by considering only rows and columns containing roots in D Hence, it is equivalent to applying the (total) orders <r, <bto these monomial matrices

on their non-zero entries

For each pair 0 < i < j
k, if τ i <b τ j , let γ i,j be the root on the row of τ i such

that γ i,j + τ j ∈ Σ+; otherwise, i.e τ j <b τ i , let ν i,j be the root on the row of τ j such

that ν i,j + τ i ∈ Σ+ For example, τ i = α m,i , τ j = α l,j , where i < j, so if α m,i <b α l,j,

i.e m < l, then γ i,j = α m,l −1 ; otherwise, if α l,j <b α m,i , i.e l < m, then ν i,j = α l,m −1

It is easy to see that ν i,j exists if and only if two hooks h(τ i ) and h(τ j) are parallel;

otherwise, γ i,j exists (Figure 1)

Let Γ D be the set of all γ i,j , let Λ D be the set of all ν i,jand let ∆D = Γ D ∪Λ D Hence,

by the definitions for the existence of γ i,j and ν i,j , Γ D ∩ Λ D=∅.

Definition 2.6 We define R =
X : γ ∈ Γ  and C =
X : ν ∈ Λ .

The next lemma provides interesting correspondences between the size of D and ∆ D,

and between w D and Γ D or Λ D Moreover, it shows that
V D , R D  = V D R D, and the

pattern subgroups R D , C D are only determined by w D in a natural way

Lemma 2.7 Let D be a basic set of size k The following are true.

(i) ∆D is closed and
X α : α ∈ ∆ D  is isomorphic to U k (q).

(ii) Γ D is closed For each pair i < j, if γ i,s , γ j,r exist and γ i,s + γ j,r ∈ Σ+, then s = j and γ i,j + γ j,r = γ i,r

(iii) Λ D is closed For each pair i < j, if ν i,s , ν j,r exist and ν i,s + ν j,r ∈ Σ+, then s = j and ν i,j + ν j,r = ν i,r

(iv) R D is isomorphic to U k (q) ∩ w D U k (q) and C D is isomorphic to U k (q) ∩ w0w D U k (q), where

w0=

⎛

⎜0 · · · 1

. .

1 · · · 0

⎞

⎟

is the longest element in S k

(v) V D R D is a pattern subgroup of U and R D normalizes V D

For example, let D = {α1,2 , α3,4, α4,5, α2,6} be a basic set in Σ+

6:

U7 (q) =

α 1,2

α 2,6

α 3,4 α4,5

and

R D=

α1,2

The next result is the main theorem, which provides a correspondence between

super-characters ξ D,φ and pattern subgroups R D

Theorem 2.8 Let ξ D,φ be a supercharacter The following are true.

(i) ξ D,φ = (λ V D R D

D )U (ii) For each χ ∈ Irr(V D R D , λ D ), χ U ∈ Irr(ξ D,φ ).

(iii) If χ1= χ2 ∈ Irr(V D R D , λ D ), then χ U = χ U

Therefore, to decompose ξ D,φ , it suffices to decompose λ V D R D

D Moreover, the induced

character λ V D R D

D is equal to

(λ D | R D

V D ∩R D)V D R D ⊗ θ, where θ is some linear character of V D R D (in Lemma 3.1) We see that λ D | R D

V D ∩R D is a

‘very special’ constituent of the regular character 1R D Hence, the decomposition method

of all supercharacters ξ D,φ of U n (q) with the same w D is generally restricted to the one

of the regular character 1R D

Here, we attempt to make a link for this special pattern R D = U k (q) ∩ w D U k (q) in Lemma 2.7 Denoting U ∩ w U by U w , where U = U n (q) and w ∈ S n is the Weyl group of

GLn (q), Thompson [16] conjectured that, for each pair r, s ∈ S n, the cardinality of the

double coset U r \ U/U s is a polynomial in q with integer coefficients In addition, U walso takes an important role when one studies GLn (q) as groups with a (B, N )-pair, such as,

for example, the Bruhat decomposition

From Theorem 2.8 and Lemma 2.7 (v), we obtain a nice decomposition of ξ D,φ

Corollary 2.9 Let ξ D,φ be a supercharacter The following are true:

(i) Irr(ξ D,φ) ={χ U : χ ∈ Irr(V D R D , λ D)};

(ii) ξ D,φ=

χ ∈Irr(V D R D ,λ D)χ(1)χ U

Theorem 2.4, Lemma 2.7 and Corollary 2.9 give a clear proof for the following corollary,

which is a different version of [1, Theorem 1.4].

Corollary 2.10.

(ξ D,φ , ξ 

D
,φ
) =

[V D R D : V D] if (D, φ) = (D  , φ  ),

3 All proofs

In this section, we prove Theorem 2.8 mainly to give a correspondence between

super-characters ξ D,φ and pattern subgroups U k (q) ∩ w D U k (q), where k = |D| First, we shall

prove Lemma 2.7

Proof of Lemma 2.7 Suppose that D = {τ1, , τ k } in ascending order <r

(i) If we rearrange D in ascending order of <b to be{θ1 , , θ k }, it is clear that, on the row of θ i, ∆D has (k − i) roots and the row of θ k does not have any root in ∆D

For each pair i < j ∈ [1, k], let ω i,j ∈ ∆ D be the root on the row of τ i such that

ω i,j + τ j ∈ Σ+ (Note that ω i,j is either γ ∈ Γ D or ν ∈ Λ D ) Hence, if τ i = α i1,i2 <bτ j =

α j1,j2 , i.e i1< j1, we have ω i,j = α i1,j1−1 Therefore, for each ω i,j = α i1,j1−1 <rω m,l=

α m1,l1−1 ∈ ∆ D , if ω i,j + ω m,l ∈ Σ+, then j1 must equal m1, and ω i,j + ω j,l = α i1,l1−1=

ω i,l This shows that ∆Dis closed, and the longest root in ∆D is ω 1,2+· · ·+ω k −1,k = ω 1,k

So ω i,j corresponds to α i,j −1 in the positive root set Σ+k −1 Therefore,
X α : α ∈ ∆ D  is

a pattern subgroup isomorphic to U (q).

(ii) With the same argument as in (i), by the definition of γ i,s and γ j,r , if γ i,s + γ j,r ∈

Σ+, then s = j By the transitive property of <rand <bon τ i , τ j , τ r , from τ i <r, <b τ j

and τ j <r, <bτ k we have τ i <r, <bτ r So γ i,r exists and γ i,j + γ j,r = γ i,r follows

(iii) The argument of (ii) holds for ν i,s and ν j,r ∈ Λ D

(iv) Let w D = (w i,j) ∈ S k ⊂ GL k (q) Since w D is a monomial matrix, w −1

D = wTD,

the transpose of w D For each X = (x i,j)∈ U k (q), we observe Y := w D · X · w −1

D Let

Y = (y i,j ) For each pair i < j, we have

y i,j =

s,r ∈[1,k]

w i,s x s,r w j,r

Since i, j are fixed, there exist unique 1
f, h
k such that w i,f = 1 = w j,h, and others

w i,s = 0 = w j,r Hence, y i,j = w i,f x f,h w j,h

Since h = f and all x s,r = 0 if r < s, we have the following:

• y i,j = 0 if f > h, i.e w i,f <b w j,h and w j,h <r w i,f;

• y i,j has non-zero value if f < h, i.e w i,f <b w j,h and w i,f <r w j,h

So R D is isomorphic to U k (q) ∩ w D U k (q) by the definition of γ i,j ∈ Γ D And, hence, C D

is isomorphic to U k (q) ∩ w0·w D U k (q) by (i)–(iii) and ∆ D = Γ D ∪ Λ D

(v) From the definition of γ i,j , it is easy to check that R D normalizes V D Hence, V D R D

Set

K D=
X α : X α ⊂ V D and α / ∈ D =
X α : X α ⊂ V D ∩ ker(λ D).

It is clear that K D is normal in V D , [V D : K D ] = q |D| and V D = K D ·τ ∈D X τ To prove Theorem 2.8, we need the following lemma

Lemma 3.1 Let ξ D,φ be a supercharacter The following are true.

(i) K D ⊂ ker(λ V D R D

D ) Moreover, λ V D R D

D (x) = [V D R D : V D ]λ D (x) for all x ∈ V D (ii) (K D ∩ R D) R D and (V D ∩ R D )/(K D ∩ R D)⊂ Z(R D /(K D ∩ R D )).

(iii) Let ¯ φ D={λ α,s ∈ φ: X α  R D } We have

λ V D R D

D = (λ D | R D

V D ∩R D)V D R D ⊗ 

λ α,s ∈ ¯ φ D

(λ α,s | V D)V D R D



.

Proof (i) It is enough to show the statement for all X α ⊂ V D By Lemma 2.7 (v)

V D  V D R D, we have

λ V D R D

D (x) = 1

|V D | y ∈V

λ D (x y)

for all x ∈ V D For each x ∈ X α , we suppose that there is X β ⊂ V D R D such that

α + β ∈ Σ+, and hence X α+β ⊂ V D We shall show that λ D (x y ) = λ D (x) for all y ∈ X β

Since X τ ∩ [V D , V D] = {1} for all τ ∈ D, we have X α+β ⊂ K D ⊂ ker(λ D) Thus,

[λ (x), λ (y)] = λ ([x, y]) = 1 since [x, y] ∈ X , i.e λ (x) −1 λ (x y) = 1

(ii) By the definition of K D  V D and V D = K D ·τ ∈D X τ, it suffices to show that

(K D ∩ R D) R D This is clear because for all X α ⊂ K D ∩ R D and all X β ⊂ R D either

α + β ∈ Σ+or X α+β ⊂ K D ∩ R D

(iii) The inflations to V D R D of λ D | R D

V D ∩R D and λ α,s | V D , for all λ α,s ∈ ¯φ D, follow directly

By Lemma 3.1 (iii), if R D ∩V D={1}, λ V D R D

D is equivalent to 1R D, the regular character

of R D In general, λ V D R D

D is equivalent to a constituent of 1R D with R D ∩ K D in the kernel Now we prove Theorem 2.8

Proof of Theorem 2.8 (i) This is clear by the transitivity of induction.

(ii) Suppose that D = {τ1 , , τ k } in ascending order <rand

λ D= 

τ i ∈D

λ τ i ,s i | V D ,

where s i ∈ F ×

q

First, we show that, for each χ ∈ Irr(V D R D , λ D ), χ U is irreducible By the transitive

property of induction, we shall induce χ from V D R D to U by a sequence of inductions along the arms of τ1, τ2, , τ k respectively by <rorder Now we setup these such induc-tion steps

For each τ i ∈ D, let A(τ i) = {α ∈ arm(τ i ) : X α  V D R D }, and c i = |A(τ i)| Let d0 = 0 and d i = d i −1 + c i for all i ∈ [1, k] Now, if c i > 0, i ∈ [1, k], we arrange A(τ i) in

decreasing order <r to be{β d i−1+1, , βd i−1 +c i } Let M0 = V D R D , M i+1 = M i
X β i

for all i ∈ [1, d k ] It is clear that M d k+1= U and X β j normalizes M j; hence, this sequence

of pattern subgroups is well defined

For each β j ∈ arm(τ i ), j ∈ [1, d k ], there exists a unique δ ∈ leg(τ i ) such that β j +δ = τ i

and X δ ⊂ K D , since if X δ  K D , there exists τ m ∈ D such that δ ∈ arm(τ m), so

τ i <r τ m , τ i <b τ m , and this implies β j = γ i,m We number this δ as δ j, and let

L(D) = {δ j : j ∈ [1, d k]} By Lemma 3.1 (i), X δ ⊂ ker(χ) for all δ ∈ L(D) Now we proceed the induction of χ from V D R D to U via a sequence of pattern subgroups along the arms of all τ i ∈ D, namely from M0 to M1, , M d k+1= U

Suppose that χ M j ∈ Irr(M j ) for some M j , j ∈ [1, d k + 1], and X δ t ⊂ ker(χ L) for all

t ∈ [j, d k ] If j = d k+ 1, the proof is complete Otherwise, the next induction step is

from M j to M j+1 = M j X β j , and we suppose that it happens on the arm of τ i For each

x ∈ X ×

β j , since [X δ j , x] = X τ i , there is some y ∈ X δ j such that λ τ i ,s i ([y, x]) = 1 and

x (χ M j )(y) = χ M j (y x ) = χ M j ([y, x]y) = λ τ i ,s i ([y, x])χ M j (y) = χ M j (y) = χ M j (1).

Hence, X δ j  ker(x (χ M j)), and

x (χ M j)= χ M j for all x ∈ X β × j

This shows that the inertia group I M j X βj (χ) = M j and χ M j X βj ∈ Irr(M j X β j , λ D)

It is easy to check directly that X δ t ⊂ ker(χ M j X βj

) for all t ∈ [j + 1, d k] by using

[X β j , X δ t] ⊂ ker(χ M j ) Therefore, we have χ U is irreducible for all χ ∈ Irr(V D R D , λ D)

by induction on j.

(iii) Now suppose χ1= χ2 ∈ Irr(V D R D , λ D ) and χ M j

1 = χ M j

2 for some M j As above,

it is enough to show that

χ M1j X βj = χ M j X βj

where β j ∈ arm(τ i) Note that

X δ j ⊂ ker(χ M j

1 )∩ ker(χ M j

2 ).

By the Mackey Formula with the double coset M j \ M j X β j /M j represented by X β j,

(χ M1 j X βj , χ M2j X βj) =

x ∈X βj

(χ M j

1 , x (χ M j

2 )).

By using the same argument as in (ii),

X δ j  ker(x (χ M j

2 )) for all x ∈ X ×

β j

Hence,x (χ M j

2 )= χ M j

1 for all x ∈ X ×

β j since X δ j ⊂ ker(χ M j

1 ) Therefore,

(χ M1j X , χ M2 j X βj ) = (χ M j

1 , χ M j

2 ) = 0, since χ M j

1 = χ M j

Note that V D R D is not normal in U In the proof of Theorem 2.8, although all induc-tions from V D R D to U are irreducible, Clifford correspondence cannot be applied The technique of a sequence of inductions from M j to M j+1 ⊂ N U (M j) has been used to control distinct induced characters

Since V D is normal in V D R D and V D R D /V D ∼ = R D /(V D ∩ R D), by Theorem 2.8 and Lemma 3.1 (iii), we only need to decompose λ D | R D

V D ∩R D instead of decomposing the

supercharacter ξ D,φ = λ U D Hence, all work is restricted to a pattern subgroup of U k (q), where k = |D| < n.

Proof of Corollary 2.9 Theorem 2.8 gives a one-to-one correspondence on the

multiplicities and degrees between Irr(V D R D , λ D ) and Irr(ξ D,φ), i.e

|Irr(V D R D , λ D)| = |Irr(ξ D,φ)|, and if χ ∈ Irr(V D R D , λ D ) has multiplicity t, then χ U ∈ Irr(ξ D,φ ) also has multiplicity t,

and

χ U (1) = [U : V D R D ]χ(1).

Therefore, it is enough to show that χ ∈ Irr(R D , λ D | V D ∩R D ) has multiplicity χ(1).

By Lemma 3.1 (i),

K D ∩ R D ⊂ ker(λ D | V D ∩R D)∩ ker(λ D | R D

V D ∩R D)

is normal in R D So λ D | V D ∩R D can be considered as a linear character of the quotient

group R D /(K D ∩ R D ) By Lemma 3.1 (ii), (V D ∩ R D )/(K D ∩ R D)⊂ Z(R D /(K D ∩ R D)),

λ D | V ∩R is a linear character of the centre and the claim holds 

Acknowledgements. The content of this paper was presented at the Conference on Algebraic Topology, Group Theory and Representation Theory, dedicated to the 60th birthdays of Professors Ron Solomon and Bob Oliver, which was held on the Isle of Skye

in 2009 The author is grateful to the organizers of the conference

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3 All proofs

In this section, we prove Theorem 2.8 mainly to give a correspondence between

super-characters ξ D,φ and pattern subgroups U k (q) ∩