James' Submodule Theorem and the Steinberg Module
James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split BN-pair. This gives rise to a distinguished composition factor o...
Gespeichert in:
| Datum: | 2017 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2017
|
| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/149266 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | James' Submodule Theorem and the Steinberg Module / M. Geck // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 10 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149266 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1492662019-02-20T01:23:40Z James' Submodule Theorem and the Steinberg Module Geck, M. James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split BN-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor. 2017 Article James' Submodule Theorem and the Steinberg Module / M. Geck // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 20C33; 20C20 DOI:10.3842/SIGMA.2017.091 http://dspace.nbuv.gov.ua/handle/123456789/149266 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split BN-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor. |
| format |
Article |
| author |
Geck, M. |
| spellingShingle |
Geck, M. James' Submodule Theorem and the Steinberg Module Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Geck, M. |
| author_sort |
Geck, M. |
| title |
James' Submodule Theorem and the Steinberg Module |
| title_short |
James' Submodule Theorem and the Steinberg Module |
| title_full |
James' Submodule Theorem and the Steinberg Module |
| title_fullStr |
James' Submodule Theorem and the Steinberg Module |
| title_full_unstemmed |
James' Submodule Theorem and the Steinberg Module |
| title_sort |
james' submodule theorem and the steinberg module |
| publisher |
Інститут математики НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149266 |
| citation_txt |
James' Submodule Theorem and the Steinberg Module / M. Geck // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 10 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT geckm jamessubmoduletheoremandthesteinbergmodule |
| first_indexed |
2025-07-12T21:11:47Z |
| last_indexed |
2025-07-12T21:11:47Z |
| _version_ |
1837477096425783296 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 091, 6 pages
James’ Submodule Theorem
and the Steinberg Module
Meinolf GECK
IAZ - Lehrstuhl für Algebra, Universität Stuttgart,
Pfaffenwaldring 57, D–70569 Stuttgart, Germany
E-mail: meinolf.geck@mathematik.uni-stuttgart.de
URL: http://www.mathematik.uni-stuttgart.de/~geckmf/
Received August 29, 2017, in final form November 28, 2017; Published online December 05, 2017
https://doi.org/10.3842/SIGMA.2017.091
Abstract. James’ submodule theorem is a fundamental result in the representation theory
of the symmetric groups and the finite general linear groups. In this note we consider
a version of that theorem for a general finite group with a split BN -pair. This gives rise
to a distinguished composition factor of the Steinberg module, first described by Hiss via
a somewhat different method. It is a major open problem to determine the dimension of
this composition factor.
Key words: groups with a BN -pair; Steinberg representation; modular representations
2010 Mathematics Subject Classification: 20C33; 20C20
1 Introduction
Let G be a finite group of Lie type and Stk be the Steinberg representation of G, defined over
an arbitrary field k; see [8]. We shall be concerned here with the case where Stk is reducible.
There is only very little general knowledge about the structure of Stk in this case; see, e.g., [3, 4]
and the references there. Using the theory of Gelfand–Graev representations of G, Hiss [5]
showed that Stk always has a certain distinguished composition factor with multiplicity 1. It
appears to be extremely difficult to determine further properties of this composition factor,
e.g., its dimension. The purpose of this note is to show that this composition factor can be
defined in a somewhat more intrinsic way through a version of James’ submodule theorem [6];
see Remark 3.4.
2 Groups with a split BN -pair
Let G be a finite group and B,N ⊆ G be subgroups which satisfy the axioms for an “algebraic
group with a split BN -pair” in [1, Section 2.5]. We just recall explicitly those properties of G
which will be important for us in the sequel. Firstly, there is a prime number p such that we
have a semidirect product decomposition B = U o H where H = B ∩ N is an abelian group
of order prime to p and U is a normal p-subgroup of B. The group H is normal in N and
W = N/H is a finite Coxeter group with a canonically defined generating set S; let l : W → N0
be the corresponding length function. For each w ∈ W , let nw ∈ N be such that Hnw = w.
Let w0 ∈ W be the unique element of maximal length; we have w2
0 = 1. Let n0 ∈ N be
a representative of w0 and V := n−10 Un0; then U ∩ V = H. For w ∈W , let Uw := U ∩ n−1w V nw.
This paper is a contribution to the Special Issue on the Representation Theory of the Symmetric Groups
and Related Topics. The full collection is available at https://www.emis.de/journals/SIGMA/symmetric-groups-
2018.html
mailto:meinolf.geck@mathematik.uni-stuttgart.de
http://www.mathematik.uni-stuttgart.de/~geckmf/
https://doi.org/10.3842/SIGMA.2017.091
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html
2 M. Geck
(Note that V , Uw do not depend on the choices of n0, nw.) Then we have the following sharp
form of the Bruhat decomposition:
G =
∐
w∈W
BnwUw, with uniqueness of expressions,
that is, every g ∈ BnwB can be uniquely written as g = bnwu where b ∈ B and u ∈ Uw.
Note that, since w2
0 = 1, we have Uw0 = U and Bn0B = Bn0U = Un0B, in both cases with
uniqueness of expressions.
Let k be any field and kG be the group algebra of G. All our kG-modules will be finite-
dimensional and left modules.
Remark 2.1. Let b :=
∑
b∈B
b ∈ kG. Then kGb is a left kG-module which is canonically
isomorphic to the induced module IndGB(kB); here, kB denotes the trivial kB-module. Now kGb
carries a natural symmetric bilinear form 〈 , 〉 : kGb× kGb→ k such that, for any g, g′ ∈ G, we
have
〈gb, g′b〉 =
{
1, if gB = g′B,
0, otherwise.
This form is non-singular and G-invariant. For any subset X ⊆ kGb, we denote
X⊥ := {a ∈ kGb | 〈a, x〉 = 0 for all x ∈ X}.
If X is a kG-submodule of kGb, then so is X⊥.
Remark 2.2. Let σ : U → k× be a group homomorphism and set
uσ =
∑
u∈U
σ(u)u ∈ kG.
Then Γσ := kGuσ is a left kG-module which is isomorphic to the induced module IndGU (kσ),
where kσ denotes the 1-dimensional kU -module corresponding to σ. Note that u2σ = |U |uσ.
Hence, if |U |1k 6= 0, then Γσ is a projective kG-module.
We say that σ is non-degenerate if the restriction of σ to Us is non-trivial for every s ∈ S.
Note that this can only occur if |U |1k 6= 0. In the case where G = GLn(q), the following result
is contained in James [6, Theorem 11.7(ii)]; see also Szechtman [9, Note 4.9, Section 10].
Lemma 2.3. Assume that |U |1k 6= 0 and σ is non-degenerate. Then the subspace uσkGb is
1-dimensional and spanned by uσn0b. Furthermore, uσnwb = 0 for all w ∈W such that w 6= w0.
Proof. By the Bruhat decomposition, we can write any g ∈ G in the form g = unwb where
u ∈ U , w ∈ W and b ∈ B. Now note that bb = b for all b ∈ B and uσu = σ(u)−1uσ for all
u ∈ U . Thus, uσkGb is spanned by {uσnwb |w ∈ W}. Now let w ∈ W be such that w 6= w0.
We shall show that uσnwb = 0. For this purpose, we use the factorisation U = UwUw0w where
Uw ∩ Uw0w = {1}; see [1, Proposition 2.5.12]. Since w 6= w0, there exists some s ∈ S such that
l(ws) > l(w) and so Us ⊆ Uw0w; see [1, Propositions 2.5.7(i) and 2.5.10(i)]. (Note that Us is
denoted by Xi in [loc. cit.].) Since nw−1 ∈ n−1w H, we obtain
uσnw−1b = uσn
−1
w b =
( ∑
u1∈Uw
σ(u1)u1
)
n−1w
( ∑
u2∈Uw0w
σ(u2)nwu2n
−1
w b
)
.
James’ Submodule Theorem and the Steinberg Module 3
By the definition of Uw0w, we have nwUw0wn
−1
w ⊆ U and so nwu2n
−1
w b = b for every fixed
u2 ∈ Uw0w. Hence, we obtain∑
u2∈Uw0w
σ(u2)nwu2n
−1
w b =
( ∑
u2∈Uw0w
σ(u2)
)
b.
Finally, since Us ⊆ Uw0w and the restriction of σ to Us is non-trivial, the above sum evaluates
to 0. Thus, uσnw−1b = 0 for all w 6= w0. Since w0 = w−10 , this also implies that uσnwb = 0 for
all w 6= w0, as required.
Hence, uσkGb is spanned by uσn0b. Finally, by the sharp form of the Bruhat decomposition,
every element of Bn0B has a unique expression of the form un0b where u ∈ U and b ∈ B. In
particular, uσn0b 6= 0 and so dim uσkGb = 1. �
Corollary 2.4. Assume that |U |1k 6= 0 and σ is non-degenerate. Then the map ϕ : Γσ → kGb,
γ 7→ γn0b, is a non-zero kG-module homomorphism and every homomorphism Γσ → kGb is
a scalar multiple of ϕ.
Proof. The fact that ϕ, as defined above, is a kG-module homomorphism is clear; it is non-
zero since ϕ(uσ) = uσn0b 6= 0 by Lemma 2.3. Since uσ is a non-zero scalar multiple of an
idempotent (see Remark 2.2), we have HomkG(Γσ, kGb) ∼= uσkGb and this is 1-dimensional by
Lemma 2.3. �
3 The submodule theorem
We keep the notation of the previous section and assume throughout that |U |1k 6= 0. For any
group homomorphism σ : U → k×, we denote by σ∗ : U → k× the group homomorphism given
by σ∗(u) = σ(u)−1 for all u ∈ U . Note that, if σ is non-degenerate, then so is σ∗. We can now
state the following version of James’s “submodule theorem” [6, Theorem 11.2], [6, 11.12(ii)].
Proposition 3.1 (cf. James [6, Theorem 11.2]). Let σ : U → k× be non-degenerate and consider
the submodule Sσ := kGuσn0b ⊆ kGb. Then the following hold.
(i) If M ⊆ kGb is any submodule, then either Sσ ⊆M or M ⊆ S ⊥
σ∗.
(ii) We have Sσ 6⊆ S ⊥
σ∗ and Sσ ∩S ⊥
σ∗ $ Sσ is the unique maximal submodule of Sσ.
(iii) The kG-module Dσ := Sσ/(Sσ ∩ S ⊥
σ∗) is absolutely irreducible and isomorphic to the
contragredient dual of Dσ∗.
(iv) Dσ occurs with multiplicity 1 as a composition factor of kGb and of Sσ.
Proof. Having established Lemma 2.3 and Corollary 2.4, this readily follows from the general
results in [6, Chapter 11]; the only difference is that James also assumes that Sσ = Sσ∗ . As
our notation and setting are somewhat different from those in [6], we recall the most important
steps of the argument.
(i) Since M ⊆ kGb, it follows from Lemma 2.3 that, for any m ∈M , there exists some cm ∈ k
such that uσm = cmuσn0b. If there exists some m ∈M with cm 6= 0, then uσn0b = c−1m uσm ∈M
and, hence, we have Sσ ⊆M in this case. Now assume that cm = 0 for all m ∈M ; that is, we
have uσM = {0}. Let m ∈ M and g ∈ G. Using the definition of uσ, uσ∗ and the G-invariance
of 〈 , 〉, we obtain
〈m, guσ∗n0b〉 =
〈
g−1m, uσ∗n0b
〉
=
〈
uσ
(
g−1m
)
, n0b
〉
= 0,
where the last equality holds since uσM = {0}. Thus, M ⊆ S ⊥
σ∗ in this case.
4 M. Geck
(ii) For u, u′ ∈ U , we have un0B = u′n0B if and only if u = u′, by the sharp form of the
Bruhat decomposition. Thus, we obtain
〈uσn0b, uσ∗n0b〉 =
∑
u,u′∈U
σ(u)σ∗(u′)〈un0b, u′n0b〉 = |U |1k 6= 0,
which means that Sσ 6⊆ S ⊥
σ∗ and so Sσ ∩ S ⊥
σ∗ $ Sσ. Now let M $ Sσ be any maximal
submodule. Then (i) immediately implies that M = Sσ ∩S ⊥
σ∗ .
(iii) By (ii), we already know that Dσ is irreducible. The remaining assertions then follow
exactly as in the proof of [6, Theorem 11.2].
(iv) By construction, Dσ occurs at least once in kGb and in Sσ. Let ϕ : Γσ → kGb be
as in Corollary 2.4. Since Γσ is projective and ϕ(Γσ) = Sσ, some indecomposable direct
summand of Γσ is a projective cover of Dσ and so the desired multiplicity of Dσ is at most
dim HomkG(Γσ, kGb) = 1, where the last equality holds by Corollary 2.4. �
We now relate the modules Sσ, Dσ in Proposition 3.1 to the Steinberg module Stk of G, as
defined in [8]. Recall that
Stk = kGe ⊆ kGb, where e :=
∑
w∈W
(−1)l(w)nwb.
We have dim Stk = |U |; a basis of Stk is given by {ue |u ∈ U}.
Proposition 3.2. We have Sσ = kGuσe ⊆ Stk. Consequently, Dσ is a composition factor (with
multiplicity 1) of Stk.
Proof. By Lemma 2.3, we have the identity
uσe =
∑
w∈W
(−1)l(w)uσnwb = (−1)l(w0)uσn0b
and so Sσ = kGuσn0b = kGuσe ⊆ kGe = Stk. The statement about Dσ then follows from
Proposition 3.1(iv). �
Remark 3.3. Gow [4, Section 2] gives an explicit formula for the restriction of the bilinear form
〈 , 〉 : kGb× kGb→ k (see Remark 2.1) to Stk. For this purpose, he first works over Z and then
rescales to obtain a non-zero form over k. One can also proceed directly, as follows. We have
〈u1e, u2e〉 = cW
(
u−11 u2
)
1k for any u1, u2 ∈ U,
where cW (u) := |{w ∈ W |n−1w unw ∈ U}| for u ∈ U . Indeed, since 〈 , 〉 is G-invariant, it is
enough to show that 〈e, ue〉 = cW (u)1k for u ∈ U . Now, we have
〈e, ue〉 =
∑
w,w′∈W
(−1)l(w)+l(w
′)〈nwb, unw′b〉 =
∑
w∈W
〈nwb, unwb〉,
where the second equality holds since nwB = unw′B if and only if w = w′. Furthermore,
nwB = unwB if and only if n−1w unw ∈ B. Since n−1w unw is a p-element, the latter condition is
equivalent to n−1w unw ∈ U . This yields the desired formula.
Remark 3.4. Since |U |1k 6= 0, the module Γσ in Remark 2.2 is projective. Also note that
Sσ = ϕ(Γσ) where ϕ is defined in Corollary 2.4. Using also Proposition 3.2, we conclude that
dim HomkG(Γσ,Stk) = dim HomkG(Γσ, kGb) = 1. So there is a unique indecomposable direct
summand PSt of Γσ such that
HomkG(PSt,Stk) 6= 0.
James’ Submodule Theorem and the Steinberg Module 5
Being projective indecomposable, PSt has a unique simple quotient whose Brauer character is
denoted by σG by Hiss [5, Section 6]. By Proposition 3.1(iv) (and its proof), we now see that σG
is the Brauer character of Dσ.
Remark 3.5. It is known that the socle of Stk is simple; see [3, 10]. We claim that this simple
socle is contained in Sσ ∩S ⊥
σ∗ , unless Stk is irreducible. Indeed, assume that Sσ ∩S ⊥
σ∗ = {0}.
Then Dσ ⊆ Stk ⊆ kGb and, hence, Dσ belongs to the Harish-Chandra series defined by the pair
(∅, kH) (notation of Hiss [5, Theorem 5.8]). By Remark 3.4 and the argument in [5, Lemma 6.2],
it then follows that [G : B]1k 6= 0. So Stk is irreducible by [8].
4 Examples
We keep the setting of the previous section. We also assume now that G is a true finite group of
Lie type, as in [1, Section 1.18]. Thus, using the notation in [loc. cit.], we have G = GF where G
is a connected reductive algebraic group G over Fp and F : G → G is an endomorphism such
that some power of F is a Frobenius map. Then the ingredients of the BN -pair in G will also
be derived from G: we have B = BF where B is an F -stable Borel subgroup of G and H = TF
where T is an F -stable maximal torus contained in B; furthermore, N = NG(T)F and U = UF
where U is the unipotent radical of B. In this setting, one can single out a certain class of
non-degenerate group homomorphisms σ : U → k×, as follows.
The commutator subgroup [U,U] is an F -stable closed connected normal subgroup of U.
We define the subgroup U∗ := [U,U]F ⊆ U ; then [U,U ] ⊆ U∗. Furthermore, we shall fix
a group homomorphism σ : U → k× which is a regular character, that is, we have U∗ ⊆ ker(σ)
and the restriction of σ to Us is non-trivial for all s ∈ S. (Such characters always exist; see
[1, Section 8.1] and [2, Definition 14.27].) Then the corresponding module Γσ = kGuσ is called
a Gelfand–Graev module for G. Let h ∈ H and σh : U → k× be defined by σh(u) := σ(h−1uh)
for u ∈ U . Then σh also is a regular character and
uσh =
∑
u∈U
σh(u)u = huσh
−1 for all h ∈ H.
Hence, right multiplication by h−1 defines an isomorphism between the corresponding Gelfand–
Graev modules Γσ and Γσh .
Remark 4.1. Let Z(G) be the center of G. Then Z(G) ⊆ H and Z(G) = Z(G)F ; see [1,
Proposition 3.6.8]. Assume now that Z(G) is connected. Then there are precisely |H/Z(G)|
regular characters and they are all conjugate under the action of H; see [1, Proposition 8.1.2].
For any h ∈ H we have
uσhn0b = huσh
−1n0b = huσn0h
−1b = huσn0b
and so Sσ = Sσh . It follows that Sσ = Sσ∗ = Sσ′ for all regular characters σ, σ′ of U ,
and we can denote this submodule simply by S0. By Proposition 3.1(iii), the simple module
D0 := S0/(S ∩S ⊥
0 ) is now self-dual. Furthermore, we have
dim S0 > |H/Z(G)|.
Indeed, we have uσn0b ∈ S0 for all regular characters σ. Since pairwise distinct group homo-
morphisms U → k× are linearly independent, the elements uσn0b (where σ runs over all regular
characters of U) form a set of |H/Z(G)| linearly independent elements in S0.
6 M. Geck
Example 4.2. Let G = GLn(q) where q is a prime power. Then our module S0 is Sλ in James’
notation [6, Definition 11.11], where λ is the partition of n with all parts equal to 1. We claim
that S0 = Stk in this case.
Indeed, by [6, Theorem 16.5], dim S0 is independent of the field k, as long as char(k) 6= p.
Since StQ is irreducible, we conclude that dim S0 = dim Stk and the claim follows. Consequently,
by Proposition 3.1(ii), D0 is the unique simple quotient of Stk. (The facts that S0 = Stk and
that this module has a unique simple quotient are also shown by Szechtman [9, Section 4].)
However, dimD0 may certainly vary as the field k varies; see the tables in [6, p. 107].
See [3, 4.14] for further examples where S0 = Stk. On the other hand, Gow [4, Section 5]
gives examples (where G = Sp4(q)) where Stk does not have a unique simple quotient, and so
S0 $ Stk. Here is a further example.
Example 4.3. Let G = Ree
(
q2
)
be the Ree group of type 2G2, where q is an odd power of
√
3.
Then G has a BN -pair of rank 1 and so [G : B] = dim Stk+1. Let k be a field of characteristic 2.
Then kGb and Stk have socle series as follows:
kGb :
kG
ϕ2
ϕ4 ϕ3 ϕ5,
ϕ2
kG
Stk :
ϕ2
ϕ4 ϕ3 ϕ5.
ϕ2
kG
Here, ϕi (i = 1, 2, 3, 4, 5) are simple kG-modules and ϕ4 is the contragredient dual of ϕ5; see
Landrock–Michler [7, Proposition 3.8(b)]. By Proposition 3.1, we have D0
∼= ϕ3 and S0 is the
uniserial submodule with composition factors kG, ϕ2, ϕ3.
References
[1] Carter R.W., Finite groups of Lie type. Conjugacy classes and complex characters, Wiley Classics Library,
John Wiley & Sons, Ltd., Chichester, 1993.
[2] Digne F., Michel J., Representations of finite groups of Lie type, London Mathematical Society Student
Texts, Vol. 21, Cambridge University Press, Cambridge, 1991.
[3] Geck M., On the modular composition factors of the Steinberg representation, J. Algebra 475 (2017),
370–391.
[4] Gow R., The Steinberg lattice of a finite Chevalley group and its modular reduction, J. London Math. Soc.
67 (2003), 593–608.
[5] Hiss G., Harish-Chandra series of Brauer characters in a finite group with a split BN -pair, J. London Math.
Soc. 48 (1993), 219–228.
[6] James G.D., Representations of general linear groups, London Mathematical Society Lecture Note Series,
Vol. 94, Cambridge University Press, Cambridge, 1984.
[7] Landrock P., Michler G.O., Principal 2-blocks of the simple groups of Ree type, Trans. Amer. Math. Soc.
260 (1980), 83–111.
[8] Steinberg R., Prime power representations of finite linear groups. II, Canad. J. Math. 9 (1957), 347–351.
[9] Szechtman F., Steinberg lattice of the general linear group and its modular reduction, J. Group Theory 14
(2011), 603–635, arXiv:0812.2232.
[10] Tinberg N.B., The Steinberg component of a finite group with a split (B,N)-pair, J. Algebra 104 (1986),
126–134.
https://doi.org/10.1017/CBO9781139172417
https://doi.org/10.1017/CBO9781139172417
https://doi.org/10.1016/j.jalgebra.2015.11.005
https://doi.org/10.1112/S0024610703004101
https://doi.org/10.1112/jlms/s2-48.2.219
https://doi.org/10.1112/jlms/s2-48.2.219
https://doi.org/10.1017/CBO9780511661921
https://doi.org/10.2307/1999877
https://doi.org/10.4153/CJM-1957-041-1
https://doi.org/10.1515/JGT.2010.067
https://arxiv.org/abs/0812.2232
https://doi.org/10.1016/0021-8693(86)90242-5
1 Introduction
2 Groups with a split BN-pair
3 The submodule theorem
4 Examples
References
|