Cyclotomic quiver Hecke algebras I

Cyclotomic quiver Hecke algebras I

Quiver Hecke algebras and categorification

Andrew Mathas

University of Sydney

Categorification, representation theory and symplectic geometry Hausdorff Research Institute for Mathematics

November 2017

Outline of lectures

1 Quiver Hecke algebras and categorification Basis theorems for quiver Hecke algebras Categorification ofU_q(g)

Categorification of highest weight modules

2 The Brundan-Kleshchev graded isomorphism theorem Seminormal forms and semisimple KLR algebras Lifting idempotents

Cellular algebras

3 The Ariki-Brundan-Kleshchev categorification theorem Dual cell modules

Graded induction and restriction The categorification theorem

4 Recent developments

Consequences of the categorification theorem Webster diagrams and tableaux

Content systems and seminormal forms

Andrew Mathas—Cyclotomic quiver Hecke algebras I 2 / 24

Generalised Cartan matrices

Let I be a (finite) indexing set

Let C = (C_ij)_i,j∈I be a symmetrizable generalised Cartan matrix:

⇐⇒ c_ii = 2, c_ij ≤0if i 6=j, c_ij =0⇐⇒c_ji =0

and DC is symmetric for a diagonal matrix D = diag(d_i|i ∈ I), ford_i >0 We assume that C is indecomposablein the sense that if ∅ 6= J (I then there exist i ∈I \J andj ∈J such thatc_ij 6=0

A Cartan datum(P,P^∨,Π,Π^∨) for C consists of:

A weight lattice P with basisfundamental weights{Λ_i |i ∈I} Dual weight lattice P^∨ = Hom(P,Z)

Simple roots Π ={α_i|i ∈I}

Simple coroots Π^∨= {hi|i ∈ I} ⊂P^∨

A pairing such thathh_i, α_ji=c_ij and hh_i,Λ_ji=δ_ij

Let h^∗= Q⊗_ZP =⇒ symmetric bilinear form onh^∗: (α_i, α_j) =d_ic_ij Let P⁺ =L

i∈INΛi be the dominant weight lattice and Q⁺ =L

i∈INα_i thepositive root lattice The positive root α=P

ia_iα_i ∈Q⁺ has heightht(α) =P

ia_i

Andrew Mathas—Cyclotomic quiver Hecke algebras I 3 / 24

Symmetrizable quivers

Type Cartan Quiver

A_e







2 −1 ··· 0

−1... ...

... ...−1 0 ··· −1 2







C_e







2 −2 ··· 0

−1... ...

... ...−1 0 ··· −1 2







+ Be,De, E6,E7,E8, F4,G2 (finite types) A_∞

A⁽¹⁾e . . .

Ce⁽¹⁾

+ Be⁽¹⁾, De⁽¹⁾, E₆⁽¹⁾,E₇⁽¹⁾,E₈⁽¹⁾, F₄⁽¹⁾,G₂⁽¹⁾ A⁽²⁾_2e ,A⁽²⁾_2e−1,D_e+1⁽²⁾ ,E₆⁽²⁾, D₄⁽³⁾

If there are eij edges

from i toj then for i 6=j c_ij =







−e_ij, if e_ij >e_ji

−1, if e_ij <e_ji

−e_ij−e_ji, otherwise

Andrew Mathas—Cyclotomic quiver Hecke algebras I 4 / 24

Graded modules and graded algebras

In these talks, gradedwill always means Z-graded

A graded module M is a module with a decomposition M =L

d∈ZM_d If m∈M_d then m ishomogeneous of degree d and we write degm=d The graded dimension of M isdim_qM =P

d(dimM_d)q^d ∈N[q,q⁻¹]

=⇒ dimM = (dimqM)_|q=1 (assuming only finitely many M_d 6=0) If s ∈Z let q^sM be the graded module that is equal to M but with the degree shifted upwards by s so that(q^sM)_d = M_d−s.

More generally, if f(q) =P

sf_sq^s ∈ N[q,q⁻¹] let f(q)M =L

s(q^sM)^⊕fs =⇒ dim_qf(q)M =f(q) dim_qM An algebra A isgraded if A=L

k∈ZA_k and A_kA_l ⊆A_k+l

A gradedA-moduleis a graded moduleM such that A_kM_d ⊆M_k+d A map f :M−→N ishomogeneous of degreed if degf(m) =d+ degm

=⇒ Hom(M,N) =L

dHom(M,N)_d,

where Hom(M,N)_d is the space of homogeneous maps of degree d In the graded category, all isomorphisms are homogeneous of degree zero

Andrew Mathas—Cyclotomic quiver Hecke algebras I 5 / 24

Quiver Hecke algebras – the Q -polynomials

Let k= L

dkd be apositively graded commutative ring

Fix polynomials(Q_ij(u,v))_i,j∈I in k[u,v]with Q_ij(u,v) =Q_ji(v,u)and Q_ij(u,v) =





 P

a,b q_abu^av^b , if i 6=j,

0, if i =j,

where q_ab =q_a,b,i,j ∈k−2(α_i,α_j)−a(α_i,α_i)−b(α_j,α_j) andq_−cij0∈ k^×₀ For 1≤m<n−1define∂Q_ijk(u,v,w) =δ_ik^Qij(u,v_w^)−Q_−u^ij(w,v) Examples

Q_ij(u,v) =



























−(u−v)² ifi j, u−v², ifi =⇒j u²−v, ifi ⇐=j u−v, ifi −→j v −u, ifi ←−j

1, ifi / j

0, ifi =j

=⇒ ∂Q_iji=























u+w−2v ifi j,

−(u+w), ifi =⇒j u+w, ifi ⇐=j

−1, ifi −→j

1, ifi ←−j

0, otherwise

Andrew Mathas—Cyclotomic quiver Hecke algebras I 6 / 24

Quiver Hecke algebras

The symmetric group Sn acts onIⁿ by place permutations:

wi = (i_w(1), . . . ,i_w(n)), forw ∈S_n andi∈ Iⁿ

For α∈ Q⁺ let I^α={i∈Iⁿ|α=α_i1+· · ·+α_in}, where n=ht(α) Definition (Khovanov-Lauda, Rouquier 2008)

The quiver Hecke algebra, or KLR algebra,Rα is the unital associative k-algebra generated by {1i|i∈I^α} ∪ {ψ_r|1≤r < n} ∪ {y_r|1≤ r ≤n} subject to the relations:

1_i1_j= δ_i,j1_i, P

i∈I^α1_i =1, ψr1_i =1_sriψr,

y_r1i =1iy_r, y_ry_t = y_ty_r, ψ_r²1i =Q_ir,ir+1(y_r,y_r+1)1i

ψ_ry_t =y_tψ_r if s 6=r,r+1, ψ_rψ_t =ψ_tψ_r if |r −t|>1 (ψryr+1−yrψr)1_i =δ_ir,ir+11_i = (yr+1ψr −ψryr)1_i

(ψ_r+1ψ_rψ_r+1−ψ_rψ_r+1ψ_r)1i =∂Q_ir,ir+1,ir+1(y_r,y_r+1,y_r+1)1i

Let Rn =L

α∈Qn⁺Rα, where Q_n⁺ ={α∈Q⁺|ht(α) =n} Importantly, Rn is graded with the grading determined by

deg1_i =0, degy_r1_i = (α_ir, α_ir), and degψ_r1_i =−(α_ir, α_ir+1)

Diagrammatic presentation for R

The elements of Rn can be described diagrammatically:

1i=

i1i2 ir in

y_r1i =

i1i2 ir in

ψ_r1i=

i1i2 ir+1ir in

If D andE are diagrams then the diagramD ◦E is zero if the residues of the strings do no match up and, when the residues coincide, D ◦E is obtained by putting D on top of E and then rescaling using isotopy

The relations become “local” operations on the diagrams that describe how to move dots and strings past crossings.

For example, the relationyr+1ψ_r1_i = (ψ_ry_r +δ_irir+1)1_i and the braid relation (in the simply laced case when e 6=2), can be written as:

= +δir ir+1 and

i

i i±1 i i±1 i i i±1 i

= ∓

A spanning set

A reduced expression for w ∈S_n is a word w =s_a1. . .s_ak with k minimal, where sa = (a,a+1)

If w =s_a1. . .s_ak is reduced set ψ_w =ψ_a1. . . ψ_ak

Warning In general,ψ_w depends on the choice of reduced expression!

Proposition

The algebra Rn is spanned by the following set of elements:

{y₁^k1. . .y_n^knψw1_i|k1, . . . ,kn ∈N,w ∈Sn andi∈ Iⁿ}

Proof By definition, Rn is spanned by all diagrams. Using the relations, we can move all of the dots to the top of the diagram, giving the result.

Andrew Mathas—Cyclotomic quiver Hecke algebras I 9 / 24

The nil Hecke algebra

Let α=nα_i, for somei ∈I =⇒ I^α={(i, . . . ,i)} (omit 1_(iⁿ₎ below)

=⇒ Rn is generated by y1, . . . ,y_n, ψ1, . . . , ψ_n−1 with relations yryt =ytyr, ψ_r²=0, ψrψr+1ψr =ψr+1ψrψr+1,

ψ_rψ_t =ψ_rψ_t if |r −s|>1,ψ_ry_t−y_sr(t)ψ_r = δ_r+1,t−δ_r,t

=⇒ the ψ_r satisfy the S_n-braid relations =⇒ ψ_w depends only on w Further, degyr = (α_i, α_i) =2d_i =−degψt

Moreover, the ψ_r’s satisfy the relations of the nil Hecke algebraNHn

=⇒ there is an action of NH_n on the polynomial ring k[x] =k[x₁, . . . ,x_n], where y_t acts as multiplication by x_t andψ_r acts as aDemazure operator

∂_rf = _x^srf−f

r−xr+1, where ^srf(x₁, . . . ,x_n) =f(x₁, . . . ,x_r+1,x_r, . . . ,x_n) Exercise Check that ∂_r²= 0, ∂_r∂_r+1∂_r =∂_r+1∂_r∂_r+1 and that

∂r∂t = ∂t∂r if |r−t|> 1

=⇒ if w =s_a1. . .s_ak then ∂_w =∂_a1. . . ∂_ak, reduced, depends only on w For w ∈S_n define the (Schubert polynomial) pw =∂_w(x₂x₃². . .x_nⁿ⁻¹)

Andrew Mathas—Cyclotomic quiver Hecke algebras I 10 / 24

Symmetric polynomials and the coinvariant algebra

Let Sym_n =k[x]^Sn be the ring of symmetric polynomials ink[x]

By the product rule, ∂_r(fg) =∂_r(f)g+f∂_r(g)

=⇒ ∂_w is aSym_n-module endomorphism ofk[x]

Theorem

The polynomial ring k[x] is a free Sym_n-module with basis{pw|w ∈ S_n}.

Sketch Letw_[1,n] be the longest element of S_n. We claimpw_[1,n] =1.

Now, w_[1,n]= s_n−1. . .s1w_[2,n], wherew_[2,n] is the longest element of S_{2,...,n}, so

pw_[1,n] =∂_n−1. . . ∂₁∂_w[2,n](x₂x₃². . .x_nⁿ⁻¹)

=∂_n−1. . . ∂1(x2. . .xn)∂w_[2,n](x3. . .x_nⁿ⁻²) =1 by induction. Now suppose that f = P

wλwpw =0, forλw ∈Sym_n and let v ∈S_n be of minimal length such thatλ_v 6=0. Applying∂_w

[1,n]v⁻¹

to f shows thatλ_v =0.

Counting graded dimensions, with x_r in degree2d_i, completes the proof

Andrew Mathas—Cyclotomic quiver Hecke algebras I 11 / 24

Quiver Hecke algebra basis theorem

Theorem (Khovanov-Lauda, Rouquier) Let α=Q_n⁺. Then Rα has basis

{y₁^k1. . .y_n^knψw1_i|kr ≥0,w ∈ Sn,i∈I^α}

SketchWhen α=nα_i, fori ∈I, the proof reduces to the nil-Hecke case The general case is similar in spirit. Using the relations you check thatRα

has a faithful polynomial representation k[x]_α= L

i∈I^αk[x_i]

where y1, . . . ,y_n act by multiplication, 1_ik[x_j] =δ_ijk[x_i] and ψ_r1_i acts on k[x_i]via

ψ_rf(x_i) =







∂_rf(x_i), if i_r = i_r+1

srf(x_i), if there is an edge from i_r to i_r+1, Q_irir+1(y_r+1,y_r)^srf(x_i), otherwise

The faithfulness of theRα action and the freeness of k[x]α implies that the elements in the spanning set are linearly independent

Andrew Mathas—Cyclotomic quiver Hecke algebras I 12 / 24

Cyclotomic quiver Hecke algebras

Fix a dominant weight Λ∈P⁺ and for each i ∈I fix a monic polynomial κ_i(u)∈k[u]of degree (h_i,Λ) of the form:

κ_i(u) =P^(hi,Λ)

d=0 k_du^(hi,Λ)−d, where k_d ∈ kd(αi,αi)

Definition (Khovanov-Lauda, Rouquier, Brundan-Stroppel, Brundan-Kleshchev)

Let Λ∈P⁺ and α∈Q_n⁺. The Cyclotomic quiver Hecke algebra, or Cyclotomic KLR algebra,Rα^Λ is the quotient ofRα by the two-sided ideal generated by {κi₁(y1)1_i|i∈I^α}. Set Rn^Λ =L

α∈Qn⁺Rα^Λ

We abuse notation and identify the elements ψ1, . . . , ψ_n−1,y1, . . . ,y_n,1_i, and ψ_w of Rn with their images inRn^Λ

Corollary

The algebra Rn^Λ is spanned by the elements

{y₁^k1. . .y_n^knψ_w1i|k_r ≥0,w ∈S_n,i∈ I^α} It is not obvious how to find a smaller spanning set

Andrew Mathas—Cyclotomic quiver Hecke algebras I 13 / 24

Finiteness of cyclotomic quiver Hecke algebras

Proposition

The algebraRn^Λ is finitely generated as ak-module

Proof It is enough to show that for anyi∈ Iⁿ there exists a monic polynomial hr(u)∈k[u]such that hr(yr)1_i =0. By definition, such a polynomial exists when r =1. Hence, by induction, it is enough to show how to constructh_r+1(u) fromh_r(u)

Case i_r 6=i_r+1: Leth_r⁰(u) be such that h⁰_r(y_r)1_sri = 0. Then

h_r⁰(yr+1)Q_irir+1(y_r,yr+1)1_i =h⁰_r(yr+1)ψ_r²1_i= ψ_rh_r(y_r)1_sriψ_r =0 Case i_r =ir+1: Letϕ_r =ψ_r(y_r−yr+1)1_i =

(yr+1−y_r)ψ_r −2 1_i.

=⇒ ϕ_rψ_r1i =−2ψ_r1i =⇒ (1+ϕ_r)²=1i

=⇒ y_r+11i = (1+ϕ_r)y_r(1+ϕ_r)1i =⇒ h_r(y_r+1)1i =0 2 Currently, bases for Rn^Λ can be found in the literature only in type A

Andrew Mathas—Cyclotomic quiver Hecke algebras I 14 / 24

The nil Hecke algebra case

Fix i ∈ I and take α= nα_i,Λ =nΛ_i andκ_i(u) =uⁿ. Then Rα^Λ is a cyclotomic quotient of the quiver Hecke algebra Rα that we considered in the nil Hecke case.

Recall that p1= x₂x₃². . .x_n−1ⁿ⁻¹. For v,w ∈ S_n define ψ_vw =ψ_v−1y2y₃². . .y_n−1ⁿ⁻¹ψ_w

=⇒ degψ_vw =d_i

n(n−1)−2`(v)−2`(w)

Proposition

The algebra Rα^Λ has graded cellular basis {ψ_vw|v,w ∈ }. In particular, Rα^Λ

has a unique irreducible module, up to grading shift

To prove this you need to explicitly describe how the yr’s act on the irreducible module after which you can show that multiplication by y₂y₃². . .y_n−1ⁿ⁻¹ sends the “bottom” basis element to the “top” basis element and so, in particular, is non-zero

Induction and restriction functors

For β∈ Q⁺ and i ∈ I define1β,1=P

j∈I^α1_ji. Define functors:

e_i :Rβ+αi-Mod−→Rβ-Mod;M 7→1_β,iM

f_i :Rβ-Mod−→Rβ+αi -Mod;N 7→Rβa+αi ⊗_RβN Proposition (Khovanov-Lauda, Rouquier)

The functors(e_i,f_i) are an adjoint pair.

In particular, these functors are exact and send projectives to projectives There are natural cyclotomic analogues of these functors:

e_i^Λ:Rβ+α^Λ _i -Mod−→Rβ^Λ-Mod;M 7→1_β,iM f_i^Λ:R_β^Λ-Mod−→R_β+α^Λ _i -Mod;N 7→R_β^Λ_a+αi ⊗_RΛ

β N Theorem (Kashiwara, Rouquier)

The functors(e_i^Λ,f_i^Λ)are an adjoint pair.

Theorem (Kang-Kashiwara, Li)

Let kbe a commutative graded ring. Then Rα^Λ is free as ak-module.

Grothendieck groups

Let A=Z[q,q⁻¹], forq an indeterminate

Let Rep(Rn^Λ)be the Grothendieck group of the finitely generated graded Rn^Λ-modules, modulo short exact sequences

So, Rep(Rn^Λ)is the A-module generated by symbols [M], as M runs over the isomorphism classes of finitely generated Rn^Λ-modules, with relations

• [qM] =q[M] (q acts as grading shift)

• [M] = [L] + [N], whenever 0−→L−→M −→N −→0is exact Similarly, let Proj(Rn^Λ)be the split, or projective, Grothendieck group of finitely generated projective Rn^Λmodules modulo direct sums Observe that e_i^Λ andf_i^Λ induce linear endomorphisms of

Rep(R^Λ) =L

n≥0Rep(Rn^Λ) and Proj(R^Λ) =L

n≥0Proj(Rn^Λ) given by e_i^Λ[M] = [e_i^ΛM] andf_i^Λ[M] = [f_i^ΛM]

Similarly, we have Grothendieck groups Rep(Rn) andProj(Rn) and the functors e_i and f_i induce endomorphisms of

Rep(R) =L

n≥0Rep(Rn) and Proj(R) =L

n≥0Proj(Rn)

Andrew Mathas—Cyclotomic quiver Hecke algebras I 17 / 24

Quantum groups

The quantum group U_q(g) associated with(C,P,P^∨,Π,Π^∨) is the unital associativeQ(q)-algebra with generators{E_i,F_i,K_i^±|i ∈I}, subject to the relations:

K_iK_j = K_jK_i, K_iK_i⁻¹=1, [E_i,F_j] =δ_ij^Ki−K

−1 i

q−q⁻¹

K_iE_jK_i⁻¹= q^dicijE_j, K_iF_jK_i⁻¹= q^−dicijF_j P

0≤c≤1−cij(−1)^cq_1−cij

c

y

iE_i^1−cij−cE_jE_i^c =0 P

0≤c≤1−cij(−1)^cq_1−cij

c

y

iF_i^1−cij−cF_jF_i^c =0 where q_i =q^di,JmKⁱ! =Qm

k=1(q^k−q^−k)/(q−q⁻¹), andq_a

b

y

i =JbKⁱ!/JaKⁱ!Jb−aKⁱ!for integers a<b,m∈N. Let U_q⁺(g) =hE_i |i ∈ Iiand U_q⁻(g) =hF_i |i ∈Ii

=⇒ There is a PBW decomposition U_q(g)∼=U_q⁻(g)⊗U_q⁰(g)⊗U_q⁺(g) Finally, the Lusztig integral form ofU_q(g)is the A-subalgebra U_A(g) ofU_q(g)generated by the quantised divided powers E_i^(k)= E_i^k/JkKⁱ! andF_i^(k) =F_i^k/JkKⁱ!for k ≥0and i ∈ I

Andrew Mathas—Cyclotomic quiver Hecke algebras I 18 / 24

Categorification of U

(g)

Theorem (Khovanov-Luda, Rouquier)

Suppose that k is a field. Then there areA-algebra isomorphisms U_A⁻(g)∼=Proj(R) and (U_A⁻(g))^∨ ∼=Rep(R)

In fact, these are isomorphisms of twisted bialgebras where the multiplication on Rep(R)and Proj(R)is induced by the convolution product: if M ∈Rα-Mod andN ∈Rβ-Mod then

M◦N =Rα+β1_α,β⊗_Rα⊗RβM ⊗N

Andrew Mathas—Cyclotomic quiver Hecke algebras I 19 / 24

Canonical bases

The Grothendieck groups Rep(R) andProj(R) come equipped with distinguished bases:

Rep(Rn) =h[D]|D self-dual irreducible Rn-modulesi

Proj(Rn) =h[P]|P self-dual indecomposable projective Rn-modulesi Warning: different dualities are used in Rep(Rn)and in Proj(Rn). We will give more precise details later

On the quantum group side, U_q⁻(g)and U_q⁻(g)^∨ also come equipped with distinguished bases: Lusztig’scanonical basis anddual canonical basis or, equivalently, Kashiwara’s upper and lower global crystal bases

Theorem (Varagnolo-Vasserot, Brundan-Stroppel, Brundan-Kleshchev, Webster)

Assume that k is a field of characteristic zero and that C is a symmetric Cartan matrix (d_i =1 for alli ∈I). Then canonical basis of U_q⁻(g)

coincides with the basis of self-dual projective indecomposable modules and the dual canonical basis coincides with the basis of self-dual irreducible modules.

Andrew Mathas—Cyclotomic quiver Hecke algebras I 20 / 24

Categorification of highest weight modules

For each dominant weight Λ∈ P⁺ there is a unique irreducible integral highest weight module L(Λ)for U_q(g).

Let v_Λ ∈L(Λ)be a highest weight vector and define L_A(Λ) =U_A(g)v_Λ and L_A(Λ)^∨ =L

L_A(Λ)^∨_µ, where L_A(Λ)^∨_µ = Hom_A(L_A(Λ)_µ,A)

Theorem (Kang-Kashiwara, Webster)

Let C be a generalised symmetrizable Cartan matrix. Then L_A(Λ)∼=M

n≥0

Proj(Rn^Λ) and L_A(Λ)^∨∼= M

n≥0

Rep(Rn^Λ)

Prior to this result, Lauda and Vazirani proved the weaker statement that the irreducible Rn^Λ-modules categorify the crystal of L(Λ)

Andrew Mathas—Cyclotomic quiver Hecke algebras I 21 / 24

Canonical bases for integrable highest weight modules

Combining the last two results proves the following:

Corollary (Varagnolo-Vasserot, Brundan-Stroppel, Brundan-Kleshchev, Webster)

Assume that k is a field of characteristic zero and that C is a symmetric Cartan matrix (d_i =1 for alli ∈I). Then:

•The canonical basis of L_A(Λ) coincides with

{[P]|self dual projective indecomposable Rn^Λ-module, n≥0}

•The dual canonical basis ofL_A(Λ)^∨ coincides with {[D]|self dual irreducible Rn^Λ-modules, n≥0}

In general, this result cannot hold for non-symmetric Cartan matrices because there are known examples where the structure constants for the canonical bases are polynomials with negative coefficients

Andrew Mathas—Cyclotomic quiver Hecke algebras I 22 / 24

Further reading I

• J. Brundan, Quiver Hecke algebras and categorification, in Advances in representation theory of algebras, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2013, 103–133.

• J. Brundan and A. Kleshchev,Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math.,178 (2009), 451–484.

• J. Brundan and C. Stroppel, Highest weight categories arising from

Khovanov’s diagram algebra III: category O, Represent. Theory,15 (2011), 170–243. arXiv:0812.1090.

• S.-J. Kang and M. Kashiwara,Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras, Invent. Math.,190(2012), 699–742.

• M. Kashiwara, Biadjointness in cyclotomic Khovanov-Lauda-Rouquier algebras, Publ. Res. Inst. Math. Sci.,48 (2012), 501–524.

• M. Khovanov and A. D. Lauda,A diagrammatic approach to

categorification of quantum groups. I, Represent. Theory, 13(2009), 309–347.

Further reading II

• ,A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc.,363 (2011), 2685–2700.

• A. D. Lauda and M. Vazirani,Crystals from categorified quantum groups, Adv. Math.,228(2011), 803–861.

• G. Li,Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of Type A, J. Algebra, 2017, accepted Oct. 2016. arXiv:1412.3747.

• R. Rouquier,2-Kac-Moody algebras, 2008, preprint.arXiv:0812.5023.

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