The degeneration of the Bruhat graph

Fix nowKan arbitrary infinite field and letλ∈H²(X,K) be an arbitrary weight.

We label the elements ofS ={1,2, . . . , n}, so that we can expressλasPn

i=1xi$i with x_i ∈K. From now on we will regard thex_i’s as indeterminate variables.

After we fix arbitrarily an ordering of the Schubert basis (or, equivalently, of the el-ements of W) the map λ^k : H^d−k(X,K) → H^d+k(X,K) can be thought of as a square matrix with number of columns equal to the number of elements of length (d−k)/2 in W. Taking the determinant we obtain a polynomial D_k(λ) = D_k(x₁, . . . , x_n). Since the field K is infinite, the existence of λ satisfying the Lefschetz property is equivalent to D_k(x₁, . . . , x_n)6= 0, for all 0< k≤n.

The polynomialsD_k(λ)appear to be hard to compute explicitly. However, it is sufficient for our purposes to compute a single term inDk(λ): its leading term in the lexicographic orderx₁ > x₂> . . . > x_n.

Definition 6.3.6. Let I be a subset of S. We say that w I-dominates v if w = w⁰w⁰⁰, v = v⁰v⁰⁰, with w⁰, v⁰ ∈ W^I, w⁰⁰, v⁰⁰ ∈ W_I and w⁰ ≥ v⁰, w⁰⁰ ≥ v⁰⁰ (≥ is the usual Bruhat order). We say that an edgew ^γ

∨

−→v isI-relevant if v I-dominates w. A path connecting wto v isI-relevant if all its edges are I-relevant.

The Bruhat order≤is compatible with the projectionW →W/WI=W^I (cf. Lemma 4.4.4), that is if v≥w thenv⁰ ≥w⁰. It follows that v I-dominates w if and only ifv ≥w andv⁰⁰≥w⁰⁰.

Lemma 6.3.7. Let v, w∈W such that v⁰ =w⁰. Then v≥wif and only if v⁰⁰ ≥w⁰⁰. Proof. Let s∈S be such that sv⁰ < v⁰. We have sv⁰ ∈ W^I by [Deo77, Lemma 3.1], thus (sv)⁰ =sv⁰. Moreover, by the Property Z 1.1.1, we havev ≥w if and only if sv≥sw, so we can easily conclude by induction on`(v⁰).

Lemma 6.3.8. Let w ^γ

∨

−→v be an edge in B_Φ. Then w ^γ

∨

−→v is I-relevant if and only if

`(v<sup>0</sup>)≤`(w⁰) + 1.

Proof. If w ^γ

∨

−→ v is I-relevant, then `(v) = `(w) + 1 and `(v<sup>00</sup>) ≥ `(w⁰⁰), so clearly

`(v<sup>0</sup>)≤`(w⁰) + 1.

Conversely, if`(v<sup>0</sup>) = `(w⁰) thenv⁰ =w⁰ because of Lemma 4.4.4. Therefore v⁰⁰ > w⁰⁰ by Lemma 6.3.7 andw ^γ

∨

−→v must be I-relevant.

It remains to consider the case `(v<sup>0</sup>) = `(w⁰) + 1, or equivalently `(v<sup>00</sup>) = `(w⁰⁰). We claim that in this case we have v⁰⁰ =w⁰⁰, whence in particular w ^γ

∨

−→v is I-relevant. The claim is proven by induction on`(v<sup>00</sup>) =`(w⁰⁰). The case`(v⁰⁰) = 0is clear.

If s∈I then, for any z ∈W we have (zs)⁰ =z⁰ and (zs)⁰⁰=z⁰⁰s. Let s∈I such that v⁰⁰s < v⁰⁰. This implies, again by the Property Z, that w≤vsor ws≤vs.

If w ≤vs < v, then w= vs. Thus we have w⁰ = (vs)⁰ =v⁰, which is a contradiction since `(v<sup>0</sup>) =`(w⁰) + 1. If ws ≤vs then ws ^s(γ)

∨

−−−→ vs is an edge in B_Φ. Since v⁰ = (vs)⁰ andw⁰= (ws)⁰ we have`((vs)<sup>0</sup>) =`((ws)⁰) + 1and`((ws)<sup>00</sup>) =`((vs)⁰⁰) =`(v⁰⁰)−1. Hence we can apply the inductive hypothesis to getw⁰⁰s=v⁰⁰s, thus w⁰⁰=v⁰⁰.

In other words, the proof of Lemma 6.3.8 shows that an edgew ^γ

∨

−→vinB_ΦisI-relevant if and only ifv⁰ =w⁰ or v⁰⁰ =w⁰⁰.

Definition 6.3.9. The I-degenerate Bruhat Graph B^I−deg_Φ is a graph having the same vertices as the Bruhat graphB_Φ. The edges inB^I−deg_Φ are theI-relevant edges inB_Φ: for anyI-relevant edgew ^γ

∨

−→v inB_Φ we put an edge w ^πI(γ

∨)

−−−−→v inB^I−deg_Φ .

In particular, in the case I =S\ {s}the edges inB^I−deg_Φ are all labeled bymαs, with m∈N>0, or by a root in Φ(I).

Example 6.3.10. Let Φ be the root system of type A2 as in the Example 6.3.3 and let I = {t}. Then ts does not I-dominate t, although ts > t in the Bruhat order. In fact, (ts)⁰⁰ = e6> t = t⁰⁰. Thus the edge t −→ tsis not {t}-relevant. The degenerate Bruhat graphB^{t}−deg_Φ is:

e

s t

st ts

sts

α β

β

α α

β α

The graph B^I−deg_Φ describes a new action^I· ofλon H^•(X,K). We say λ^I·P_w = X

w−→v∈B^{δ I−deg}_Φ

λ(δ)P_v

where the sum runs over all edges w −→^δ v starting in w in B^I−deg_Φ (or equivalently all I-relevant edges starting inw inB_Φ). We call it theI-degenerate action of λ.

The new graph B^I−deg_Φ can be obtained as product of two smaller graphs. In fact, we have B^I−deg_Φ ∼= B^I_Φ×B_Φ(I): at the level of vertices we have a bijection W = W^I ×WI

and, because of Lemma 6.3.8, for anyI-relevant edgew ^γ

∨

−→v we have two cases:

• w⁰=v⁰ andw⁰⁰t_γ =v⁰⁰, sow ^πI(γ

∨)

−−−−→v comes from the edge w^{00 γ}

∨

−→v⁰⁰ inB_Φ(I);

• w⁰⁰ =v⁰⁰ and w⁰t_w⁰⁰_(γ) = v⁰, so w ^πI(γ

∨)

−−−−→v comes from the edgew^{0 πI(w}

00(γ)^∨)

−−−−−−−→ v⁰ in B^I_Φ.

Remark 6.3.11. It is not hard to see that the I-degenerate action described by B^I−deg_Φ coincides with the action onH^•(G/P_I×P_I/B,K)∼=H^•(G/P_I,K)⊗H^•(P_I/B,K)defined as follows: ifλ=P

i∈Sxi$i,P1 ∈H^•(G/PI,K)and P2∈H^•(PI/B,K) then λ^I·(P1⊗P2) =

X

i∈S\I

xi$i

·P1⊗P2+P1⊗ X

i∈I

xi$i

·P2.

For a polynomial f ∈ K[x₁, . . . , x_n] we denote by deg_i(f) its degree in the vari-able xi and by coeffi,a(f) the coefficient of x^a_i in f (thus coeffi,a(f) is an element of K[x1, . . . xi−1, xi+1, . . . xn]). We set deg_i(0) =−1.

Recall that the elements of S are labeled as {1,2, . . . , n} and that λ = P

ix_i$_i is a formal linear combination of the fundamental weights. We setI =S\ {1}.

We have

deg₁(λ(γ^∨)) =

(1 if γ ∈Φ\Φ(I) 0 if γ ∈Φ(I) . Notice thatγ ∈Φ(I) if and only iftγ∈WI.

Lemma 6.3.12. Let w, v ∈W with `(v)> `(w). Then:

i) deg₁(C_w,v(λ)) ≤ `(v<sup>0</sup>)−`(w⁰) and we have equality if and only if there exists an I -relevant path connectingw to v;

ii) coeff<sub>1,`(v</sub><sup>0</sup>)−`(w⁰)(Cw,v(λ))·x^{`(v</sup><sub>1</sub> <sup>0)−`(w0)}= X

relevant

λ(πI(γ₁^∨))λ(πI(γ₂^∨)). . . λ(πI(γ_k^∨)), where the sum runs over all the I-relevant paths w ^γ

∨

−→1 w1 γ₂^∨

−→ w2 γ₃^∨

−→ . . . ^γ

∨

−→k v connecting w tov in B_Φ.

Proof. i)We start with the case`(v) =`(w) + 1. If there are no edges connectingw to v inB_Φ then there is nothing to show.

Assume that there is an edge w ^γ

∨

−→ v in B_Φ, so that Cw,v(λ) =λ(γ^∨). If w ^γ

∨

−→ v is notI-relevant by Lemma 6.3.8 we have`(v<sup>0</sup>)−`(w⁰)≥2, and the statement follows since deg₁(C_w,v(λ))≤1.

Assume now thatw ^γ

∨

−→v isI-relevant, then w⁰ =v⁰ or w⁰⁰=v⁰⁰. Since w⁰w⁰⁰t_γ=v⁰v⁰⁰ we see thatw⁰ =v⁰ if and only if tγ ∈WI, i.e. if and only ifdeg₁(Cw,v(λ)) = 0.

The general case`(v)> `(w) + 1follows since Cw,v(λ) =X

Cw,w1(λ)Cw1,w2(λ). . . Cwk−1,v(λ) where the sum runs over all pathsw−→w₁ −→w₂−→. . .−→v inB_Φ.

ii) We start with the case `(v) = `(w) + 1. If there are no I-relevant edges in B_Φ between w and v then both sides are 0. If there is an I-relevant edge w ^γ

∨

−→ v, then C_w,v(λ) =λ(γ^∨) and

coeff_{1,`(v</sub><sup>0</sup><sub>)−`(w}⁰₎(λ(γ^∨))·x^{`(v</sup><sub>1</sub> <sup>0)−`(w0)}=λ(π_I(γ^∨)).

The general case`(v)> `(w) + 1easily follows.

We fix now an arbitraryk∈ {1,2, . . . , d}. LetD⁽¹⁾_k (λ) be the Lefschetz determinant of theI-degenerate action of λonH^d−k(X,K), described by B^I−deg_Φ , computed in the same basis used for D_k(λ). In other words D⁽¹⁾_k (λ) is the determinant of the map λ^{k I}· (−) : H^d−k(G/P_I×P_I/B,K)→H^d+k(G/P_I×P_I/B,K) described above.

Lemma 6.3.13. Let Mk= X

`(v)=(d+k)/2

l(v⁰)− X

`(w)=(d−k)/2

l(w⁰). Then we have:

i) deg₁(D_k(λ))≤M_k;

ii) The polynomial D_k⁽¹⁾(λ) is homogeneous of degreeM_k inx1; iii) coeff1,Mk(Dk(λ))·x^M₁ ^k =D_k⁽¹⁾(λ).

Proof. The determinant polynomial can be expressed as Dk(λ) =X

σ

sgn(σ)C_w1,σ(w1)(λ)C_w2,σ(w2)(λ). . . C_{wn(k),σ(wn(k))}(λ)

whereσ runs over all possible bijections between elements in W of length (d−k)/2 and (d+k)/2(and the sign is determined by the chosen order of the Schubert basis). Then i) follows from Lemma 6.3.12.

The terms in the sum which contribute to coeff1,M_k(D_k(λ)) are precisely the ones coming from I-relevant paths, i.e. the one which are also in D_k⁽¹⁾(λ), so ii) and iii) also follow.

We can now reiterate this procedure. Let S = I₀ ⊃ I₁ ⊃ I₂ ⊃ . . . ⊃I_n = ∅ be such thatIj−1\Ij ={j}for any 1≤j≤n. We have a length preserving bijection of sets:

Ψ :W ∼=W^I1×W_I^I2

1 ×. . .×W_In−1. We write Ψ(w) = w⁽¹⁾, w⁽²⁾, . . . , w⁽ⁿ⁾

. The degenerated graph B⁽¹⁾_Φ := B^I_Φ^1−deg is iso-morphic to B^I_Φ¹ ×B_Φ(I1). It can be degenerated again into B⁽²⁾_Φ := B^I_Φ¹ ×B^I_Φ(I^2-deg

1) ∼= B^I_Φ¹×B^I_Φ(I²

1)×B_Φ(I2), and so on up to B⁽ⁿ⁻¹⁾_Φ :=B^I_Φ¹ ×B^I_Φ(I²

1)×. . .×B_Φ(In−1). We setB⁽⁰⁾_Φ :=B_Φ andB⁽ⁿ⁾_Φ :=B⁽ⁿ⁻¹⁾_Φ .

Definition 6.3.14. Each of the B^(j)_Φ describes a new action of λ on H^•(X,K), which we call the j^th-degenerate action and we denote by ^j·. We say that v j-dominates w if v⁽ⁱ⁾ ≥w⁽ⁱ⁾ for any i≤j andv^(j+1). . . v⁽ⁿ⁾≥w^(j+1). . . w⁽ⁿ⁾.

We say that an edgew ^γ

∨

−→v isj-relevant if v j-dominatesw. A path connectingwto v isj-relevant if all its edges arej-relevant.

For1 ≤j ≤n, let Cw,v^(j)(λ) be the coefficient of P_v in λ^{h j}·P_w, where`(v)−`(w) =h.

Thus Lemma 6.3.12.ii can be restated as:

coeff<sub>1,`(v</sub>(1))−`(w⁽¹⁾)(C_w,v⁽⁰⁾(λ))·x^{`(v</sup><sub>1</sub> <sup>(1))−`(w(1))}=C_w,v⁽¹⁾(λ).

We also have:

Lemma 6.3.15. Let w, v ∈W with `(v)> `(w) and 0≤j≤n−1. Then:

i) deg_j+1C_w,v^(j)(λ)≤`(v<sup>(j+1)</sup>)−`(w^(j+1)) and the equality holds if and only if there is a (j+ 1)-relevant path connecting v and w;

ii) coeff<sub>j+1,`(v</sub>(j+1))−`(w^(j+1))(C_w,v^(j)(λ))·x^{`(v</sup><sub>j+1</sub><sup>(j+1))−`(w(j+1))}=C_w,v^(j+1)(λ);

iii) Cw,v^(j+1)(λ), regarded as a polynomial in x_i, is homogeneous of degree `(v<sup>(i)</sup>)−`(w⁽ⁱ⁾) for 1≤i≤j+ 1.

Proof. The same arguments as in the proof of Lemma 6.3.12 show (i) and (ii). Now (iii) follows by induction onj using (ii).

For 0 ≤ j ≤ n let D^(j)_k (λ) be the Lefschetz determinant obtained from the j^th -degenerate action ofλ, computed in the same bases used forD_k(λ). We have

D^(j)_k (λ) =X

σ

sgn(σ)C_w^(j)

1,σ(w1)(λ)C_w^(j)

2,σ(w2)(λ). . . C_w^(j)

n(k),σ(wn(k))(λ). (6.3) For any 1≤j≤n letM_k^(j)= X

`(v)=^d+k₂

`(v^(j))− X

`(w)=^d−k₂

`(w^(j)).

Lemma 6.3.16. For any 0≤j≤n−1 we have:

i) deg_j+1D^(j)_k (λ)≤M_k^(j+1);

ii) D_k^(j)(λ) is homogeneous of degree M_k⁽ⁱ⁾ in x_i for 1≤i≤j;

iii) coeff

j+1,M_k^(j+1)D_k^(j)(λ)·x^M

(j+1) k

j+1 =D^(j+1)_k (λ).

Proof. Using (6.3) and Lemma 6.3.15 this follows arguing just as in Lemma 6.3.13.

Letµ_k=x^M

(1) k

1 x^M

(2) k

2 ·. . .·x^M

(n)

nk . We have the following:

Corollary 6.3.17. All monomials in Dk(λ) =Dk(x1, . . . , xn) are smaller thanµk in the lexicographic order.

The polynomial D⁽ⁿ⁻¹⁾_k (λ) (which is equal to D_k⁽ⁿ⁾(λ)) is homogeneous of degree M_k^(j) in xj for any 1 ≤j ≤n, i.e. D_k⁽ⁿ⁻¹⁾(λ) = Rkµk, with Rk ∈K, and the coefficient of the monomialµ_k in D_k(λ) isR_k.

Im Dokument Hodge theoretic aspects of Soergel bimodules and representation theory (Seite 103-108)