Character formulae and tensor ideals

This chapter presents the character formulae for indecomposable quantum tilting mod-ules. The characters are expressed by certain basis change coefficients in a Hecke algebra module. We introduce this module, the Hecke module, first and then go on to state So-ergel’s character formula.

This chapter contains also a classification of tensor ideals of quantum tilting modules.

The proof relies on Hecke algebra combinatorics, and may be seen as an application of the right cell theory of Chapter 4 and of the character formula for indecomposable quantum tilting modules. We conclude the chapter by introducing weight cells.

Assumep hthroughout this chapter.

The Hecke module

The material in this section may be found in (Soergel 1997).

The finite Weyl groupW0is a parabolic subgroup ofW, so each right coset ofW0 W has a unique representative of minimal length. We denote the set of these representatives byW⁰; multiplication then gives a bijectionW0 W⁰ W. The setW⁰is described thoroughly by Lemma 4.19. In particular, if we associate to each element x W⁰ the alcovexCas in figure 2,3, and 4 then we get all alcoves in the dominant Weyl chamber.

LetH₀denote the Hecke algebra ofS0W0. It is a subalgebra ofH. We have a surjec-tive vv ¹-algebra homomorphism,φ

v:H₀

vv ¹, mapping each generatorHs_i, si S0, to v. This gives vv ¹ aH₀-module structure, and by induction we obtain a rightH-module

N _vv ¹ H₀H

This module is the Hecke module. In Lemma 5.5 below we show that N may also be constructed as a quotient ofH with a right cell ideal.

We denote byεthe canonical surjection

ε:H N

H 1 H

Note that forz W0,x W⁰ we haveεHzx v^lzNx. N has a basis consisting of

Nx 1 Hx x W⁰ and the action ofH is given by

NxH_s

Nxs^! vNx xs^" xandxs W⁰; Nxs^! v ¹Nx; xs^# x;

0 xs^$ W⁰;

(5.1)

The first and second line follows from equation (4.3). The assumption in the last line forces xs txfor somet S0, which gives the formula.

We define an involution onN _{bya H a} H. This involution isH-skewlinear (i.e.

NH N HforN NH H). We say thatNis selfdual ifN N. As was the case with H, we want to replace the canonical basis Nx 1 Hx x W⁰ with one of selfdual elements.

35

36 5. CHARACTER FORMULAE AND TENSOR IDEALS

THEOREM5.1. For each x W⁰there is a unique selfdual element N_xinN _{so that} N_x Nx

∑

y xvvNy

PROOF. This theorem is proved by the same method as Theorem 4.1. In particular, the proof is constructive and gives an algorithm to obtain the selfdual elementN_x.

To begin, note thatNe 1 Heis selfdual; this shows the existence part of the theorem forxof length 0.

Givenxof length 1 we chooses Sso thatxs x. By induction we may assume the existence of the selfdualN_xs Nxs ∑y xsnyxsNy, withnyxs vv. ThenN_xsH_sis selfdual. We write

N_xsH_s Nx

∑

y x

nyNy (5.2)

wherenyv by (5.1). So constant coefficients may appear inN_xsH_s. We fix this with N_x N_xsH_s

∑

y xny0 N_y which is selfdual, and this shows the existence part forx.

We leave uniqueness to the reader.

L^EMMA5.2.

εH_x

N_x x W⁰ 0 x W⁰

PROOF. The first line follows from the fact thatεH_x is selfdual andεH_x Nx

∑y xvvεHy Nx ∑z xvvNz. For the second line, note thatεH_si εHs_i v 0 forsi s0. Whenx W⁰there is asi s0so thatx six. Therefore we may prove the

second line by induction via Remark 4.4.

DEFINITION5.3. Let nyx N_x:Nyv denote the coefficients of the new selfdual basis expressed in the old basis, so that

N_x

∑

y xnyxNy

REMARK5.4. The mapεreveals that the polynomials nyxand hyxare related by the formula

nyx

∑

w W₀

v^lwhwyx (5.3)

Recall that we denote the linear coefficient of hyxby µyx . The equation (5.3) shows that the linear coefficient of nyxis equal to µyx for xy W⁰.

The Hecke algebra acts on the selfdual basis by the following formulae; to see this use εon the equations in Proposition 4.3.

N_xH_s ^!"

""""

#"""""$

N_xs

∑

y^% _Rxy W⁰

µyxN_y xs xandxs W⁰;

y^% R

∑

xy W⁰

µyx N_y xs xandxs W⁰;

v v^{& 1}N_x xs x

(5.4)

We conclude this section with an alternative description ofN. Recall the right cell idealsI_xfrom Chapter 4.

LEMMA5.5. εinduces an isomorphism of rightH-modules betweenH _∑i

'

( 0I_siand N_.

PROOF. We have∑i^(' 0I_si*) x⁺W⁰vv^{& 1}H_xand this is the kernel ofεby Lemma

5.2.

SOERGELS THEOREM 37

We will need the following analogue of Lemma 4.5.

L^EMMA5.6. When xs x and ys y we have nysx vnyx.

Soergels Theorem

Our main interest in the Hecke module is Theorem 5.7 below. It relates the structure of quantum tilting modules with combinatorics of the corresponding Hecke algebra. Recall that for a moduleQwith a Weyl filtration, Q:Vqλ denotes the number of timesVqλ appears as a quotient in the filtration. Clearly, the numbers Q:Vqλ for allλdetermine the character ofQ. WhenQis a quantum tilting module we letQ:Tqλ denote the num-ber of timesTqλ appears as a summand ofQ; we say that Q:Tqλ is the multiplicity ofTqλ inQ.

THEOREM5.7. Let p h. For a weightλ C and xyz W⁰we have (i) Tqxλ :Vqyλ nyx1

(ii) If xs x then ΘsTqxλ :Tqzλ N_xH_s:N_z.

The proof of (i) may be found in (Soergel 1998). This result of Soergel relies on an equivalence of categories between affine Lie algebra modules and quantum group modules established in (Kazhdan and Lusztig 1993), (Kazhdan and Lusztig 1994) together with results from (Lusztig 1994) and (Kashiwara and Tanisaki 1996). In some types these results impose mild restrictions onp.

We will use the second part of Theorem 5.7 extensively. It is a straightforward conse-quence of the first part as shown in the proof below.

PROOF OF(II). Consider a moduleQwith character chQ ∑y;yλ X a_yλχyλ. Ap-plying the wallcrossing functor we obtain a module with character given by chΘsQ

∑y;yλ X a_yλχyλ χysλ.

Now consider a Weyl factorVqyλ inTqxλ. Suppose first thatys W⁰; then by the first part of the theorem,

nyx1 nysx1 Tqxλ :VqyλTqxλ :Vqysλ

ΘsTqxλ :Vqyλ

∑

z xs

ΘsTqxλ :TqzλTqzλ :Vqyλ

∑

z xs

ΘsTqxλ :Tqzλnyz1 In the first equation below, see Equation (5.1).

N_xH_{s y;}

∑

_ys

y ysW⁰

nyxNys vnyxNy nysxNy v ¹nyxNys

N_xH_s

∑

z xs

N_xH_s:N_z

∑

y;ys ynyzNy nyszNys

Considering the coefficientN_xH_s:Ny and evaluating inv 1, we find that nyx1 nysx1

∑

z xs

N_xH_s:N_znyz1

Ifys W⁰thenΘsTqxλ :Vqyλ 0 N_xH_s:Ny. So for allyz xswe have

z

∑

xs

ΘsTqxλ :Tqzλnyz1

∑

z xs

N_xH_s:N_znyz1

It is now a matter of linear algebra to conclude thatΘsTqxλ :Tqzλ N_xH_s:N_z

for allz xs.

38 5. CHARACTER FORMULAE AND TENSOR IDEALS

The Hecke module atv 1

We continue this chapter with a construction of tensor ideals of quantum tilting mod-ules. This is based on the paper (Ostrik 1997). It illustrates the strength of right cells when applied to the representation theory of quantum groups at a root of unity, through Theorem 5.7. This section prepares the ground for Ostriks tensor ideals in the next section.

Inspired by Theorem 5.7 we begin with the specialization ofN _atv 1. Consider as anvv ¹-module withvacting as multiplication by 1. We then obtain a rightH-module by

N¹ _v

v ¹ N From thevv ¹-bases ofN we get two bases ofN¹:

N¹_x 1 Nx x W⁰ N¹_x 1 N_x x W⁰

As endomorphisms ofN¹we haveHxHy Hxyeven whenlx ly lxy. Further, the generators act by the formulae below; the first equation follows from the defining relations (4.1) and (4.2) ofH and the second equation follow from equation (5.4).

N_x¹Hs

N_xs¹ ifxs W⁰;

N¹_x ifxs W⁰; (5.5)

N¹_xH_s !

N¹_xs

∑

y^" _Rx y^# W⁰

µyxN¹_y xs^$ x xs W⁰;

y^" _R

∑

x y^# W⁰

µyxN¹_y xs^$ x xs W⁰;

2N_x xs x^%

(5.6)

A right cell submodule ofN¹ is a right H-submodule with a basis consisting of

N_y y Y for some subsetY ofW. The following lemma describes all of them.

L^EMMA5.8. LetC ^& _W⁰denote a right cell, and let x C_{. Then} JC ^('

y⁾ _RC y^# W⁰

*vv ¹N¹_y

is a right cell submodule andJC is the smallest right cell submodule to contain N¹_x. A right cell submodule is a sum of suchJC, and a sum of suchJC is a right cell submodule.

REMARK 5.9. Note that the set y W⁰ y⁺ R C is a union of right cells; it is equal to the union of allC^,-+ R C.

PROOF OFLEMMA5.8. Equation (5.6) shows thatJ_C is stable under the generators H_sofH_{. So each}J_C is a right cell submodule.

Suppose that y W⁰ andy⁺ R C. We will show thatN¹_y belongs to any right cell submodule containingN¹_x. There is a chainy w0^. R w1^/0/1/. R wn x, wherewi W to begin with. But in factwi W⁰, asL_wi^2& L_y³ _s0 by Lemma 4.13. Nowwi^. R wi⁴ 1means thatH_wi

5 1H_s:H_wi⁷⁶ 0 for somes; further

H_wi

5 1H_s:H_wi N¹_wi

5 1H_s:N¹_wi

by equation (5.6). This shows that each N¹_wi must be part of any basis of a right cell submodule containingN¹_x. We have shown thatJ_C is the smallest right cell submodule containingN¹_x.

We leave the remaining assertions to the reader. ⁸

OSTRIK’S TENSOR IDEALS 39

Ostrik’s tensor ideals

Inside the categoryC_qof finite dimensionalU_q-modules we have the full subcategory of tilting modulesT_q.

D^EFINITION5.10. A subsetτ T_qis a tensor ideal if it satisfies the following condi-tions.

(i) For any Q1inτand any tilting module Q, the tensor product Q1 Q belongs to τ.

(ii) If Q is a summand in Q1and Q1belongs toτ, then Q belongs toτ.

These two properties are sufficient to ensure that tensor ideals are stable under the translation functors.

Consider a tensor idealτ T_q; it follows from the defining properties thatτis spanned by a set of indecomposable tilting modules Tqλ λ Λ (Λdenoting a subset of the dominant weights) in the sense that all modules inτare direct sums of these indecompos-ables. What can we say aboutΛ? What are the possible subsetsΛ X ?

L^EMMA5.11. Letτ T_qdenote a tensor ideal. Then Tqx0 belongs toτif and only if Tqxλ belongs toτwhenever xλis a dominant weight in the lower closure of the alcove xC.

P^ROOF. Recall that tensor ideals are stable under the translation functors. The claim follows from (Soergel 1997, Remark 7.2 2.): Whenxλbelongs to the lower closure of the alcovexC, we have

T₀^λTqx0 Tqxλ^Wλ T_λ⁰Tqxλ Tqx0

HereW_λdenotes the stabilizer ofλ.

It follows that all tensor ideals are spanned by sets Tqλ λ Λ whereΛis the union of the set of dominant weights in lower closures of alcoves.

COROLLARY5.12. To each tensor idealτinT_qwe may associate a subsetCτ W⁰ so that the indecomposable tilting modules inτare all Tqλ withλin the lower closure of one of the alcoves yC, y Cτ.

P^ROPOSITION5.13. Letτdenote a tensor ideal inT_{qand let}Cτ W⁰denote the associated set of Weyl group elements. Then x Cτ N¹_x is a right cell submodule inN¹_, andCτ is a union of right cells in W⁰.

PROOF. Letx Cτ and assume thatN¹_xH_s:N¹_y

0. We must show thaty Cτ, as this demonstrates that _x Cτ N¹_xis stable under the generatorsH_sofH. By definition ofCτ we know thatTqx0 belongs to the tensor idealτ. To show the proposition we need only verify thatTqy0 belongs toτ. This is immediate from

ΘsTqx0 :Tqy0 N_xH_s:N_y N¹_xH_s:N¹_y

The first equality above follows from Theorem 5.7 and the second by comparing the equa-tions (5.4) and (5.6). The last assertion in the proposition follows from Lemma 5.8 and

Remark 5.9.

By now we have seen that a tensor ideal inT_qcorresponds to a right cell submodule.

Further the right cell submodules separate tensor ideals, in the sense that two tensor ideals are different if and only if they correspond to different right cell submodules. This takes us some way toward a classification of tensor ideals inT_qas Lemma 5.8 provide us with a classification of right cell submodules in terms of right cells.

It remains to construct a tensor ideal from each right cell submodule. It suffices to do this for the minimal right cell submodulesJC. This occupies the rest of this section.

40 5. CHARACTER FORMULAE AND TENSOR IDEALS

We begin by connecting the ring of characters with the moduleN¹. Recall that the characters of the Weyl modules χλ λ X form a basis of the character ring. Then set

α0: X^W0 N¹ χλ

0 ifλ W0

N_x¹ ifλ x0 for somex W⁰ REMARK5.14.

(i) This map is well defined as0is a regular weight when p h.

(ii) By Theorem 5.7 we haveα0Tqx0 N¹_x. Note that we should really write α0chTqx0, but this should not cause confusion.

The following lemmata give two important properties of the mapα0. The first lemma is almost trivial, but the second is rather technical.

L^EMMA5.15. Let w W . Then

α0χw0 α0χ0Hw

P^ROOF. Decomposew w0xwithw0 W0andx W⁰satisfyinglw lw0 lx . Then, using formula (5.5) in the second line

α0χw0 α0 1^lw0χx0 ^! 1^lw0N_x¹

α0χ0Hw N_e¹Hw₀x ^! 1^lw0N_e¹Hx ^! 1^lw0N_x¹ "

The following lemma from (Ostrik 1997) is the key result needed in the proof of Proposition 5.18; there we show that a minimal right cell submoduleJ_C corresponds to a tensor ideal.

LEMMA5.16. Letλ^#µ C and assume that zλis a dominant weight. For any module Q inC_q, there is an HQ H so that

α0T_µ⁰Tqzλ^%$ Q α0T_λ⁰TqzλHQ

P^ROOF. Write chTqzλ^& ∑w^' Wawχwλ. Then by (Jantzen 1987, II.7.5b) we have chpr_µTqzλ^($ Q

∑

w^'W

aw

∑

ν⁾ν

λ^' W^*µ

dimQνχw⁺λ ν

Consider the following set of Weyl group elements:

P^, x W ν λ xµ^# dimQν^- 0

This set is clearly stable under multiplication from the right withw1 Wµ. It is also stable under multiplication from the left byw2 Wλ: We havew2xµ w2^.λ ν^& w2λ w^/ν for somew^/ in the finite Weyl group, and dimQν dimQ_w0ν.

The actions of the groupsWµ(from the right) andW_λ(from the left) are simply transi-tively. The orbits of theWµ-action are indexed by the weights ofQ, so we choose for each weightνofQan elementwνso thatwνWµµ λ ν. Letwⁱ,i 1^#1#ldenote a set of representatives from the orbits of theW_λ-action. ThenPmay be described as

P^, wνw1 w1 Wµ^# dimQν^- 0^2, w2wⁱ w2 Wλ^# i 1^#1#l³ We may now continue the calculation. We have

chpr_µTqzλ^($ Q

∑

w^' W

aw

∑

ν⁾ν

λ⁴ w_ν^*µ

dimQν χwwνµ⁵

This gives

chT_µ⁰Tqzλ^($ Q

∑

w^' W

aw

∑

ν⁾ν λ⁴ w_ν^*µ

dimQν

∑

w₁^' W_µ

χwwνw10⁵

OSTRIK’S TENSOR IDEALS 41

We use Lemma 5.15 in the following calculation.

α0 T_µ⁰Tq zλ Q

∑

wW

aw

∑

νν λ w_νµ

dim Qν

∑

w₁ W_µ

α0χ wwνw10

α0

∑

w Wawχ w0

∑

νν λ w_νµ

dim Qν

∑

w₁ W_µHw_νw₁

α0

∑

w W

awχ w0

∑

i

∑

w₂ W_λ

dim QνH_w2wi

α0

∑

w₂ W_λ

∑

w W

awχ ww20

∑

i

dim Qν H_w

i

α0 T⁰_λTq zλ

∑

i dim QνH_wi

Here the last line follows as

chT_λ⁰Tq zλ

∑

w W

aw

∑

w₂ W_λ

χ ww20

REMARK5.17. In Lemma 5.16 we may replace Tq zλ by any other module Q with pr_λQ Q.

Proposition 5.13 and Corollary 5.12 states that each tensor ideal inT_qdetermines a right cell submodule in N¹, and that two different tensor ideals are mapped to different right cell submodules. In the other direction, Proposition 5.18 shows that each of the minimal right cell modulesJC from Lemma 5.8 corresponds to a tensor ideal.

P^ROPOSITION 5.18. LetC _W⁰ denote a right cell. The indecomposable tilting modules with highest weight in the lower closure of an alcove xC, x R C span a tensor ideal.

PROOF. Letτ C denote the set of all indecomposable tilting modules with highest weight in the lower closure of one of the alcovesxC,x R C. Recall the mapα0from the ring of characters toN¹defined above. We first show that the mapα0reveals whether a given indecomposable tilting module belongs toτ C or not. Suppose thatTq yµ is such a module, withµ Candychosen so thatyµbelongs to the lower closure of the alcove yC. Then

α0 T_µ⁰Tq yµ α0 Tq y0 N¹_y

We see thatTq yµ belongsτ C if and only if it is mapped toJ_C N¹_{byα0 Tµ}⁰

. Now assume thatx R C_{. LetQ}denote an arbitrary tilting module and consider the tensor productTq xλ Q. We must show that each summandTq yµ belongs toτ C_. By the discussion above this is equivalent toα0 T_µ⁰ Tq xλ Q J_{C for allµ C. Now} Lemma 5.16 shows that

α0 T_µ⁰ Tq xλ Q α0 T_λ⁰Tq xλH

for some H H_{. As} α0 T_λ⁰Tq xλH N¹_xH and asJC is a H-module we find that

α0 T_µ⁰ Tq xλ Q JC as needed. We are done.

Denote byτ C the tensor ideal corresponding to the right cell C_{, i.e. letτ} C de-note the tensor ideal spanned by all indecomposable quantum tilting modules with highest weight in the lower closure of an alcovexCwithx R C. We may now classify the tensor ideals inT_q.

THEOREM5.19. LetC _W⁰denote a right cell and let x C. The tensor idealτ C is the smallest tensor ideal to contain Tq x0. A tensor ideal is a union of suchτ C_{, and} a union of suchτ C is a tensor ideal.

42 5. CHARACTER FORMULAE AND TENSOR IDEALS

PROOF. Let us consider the smallest tensor ideal containingTq x0. By Corollary 5.12 and Proposition 5.13 we may associate to it a subsetY W⁰, so that y Y N¹_yis a right cell submodule ofN¹. This submodule containsN¹_x, so it contains alsoJC, where C W⁰is the right cell containingx. We have shown thatC Y. Hence any tensor ideal containingTq x0 must containτ C. Further,τ C is a tensor ideal by Proposition 5.18;

we see thatτ C is the smallest tensor ideal to containTq x0.

The last assertions are clear.

Weight cells

In the previous sections we have seen that there is a close connection between right cell ideals inH and right cell submodules ofN¹on one side and tensor ideals of quantum tilting modules on the other side. Weyl group elements that generate the same right cell ideal or right cell submodule are said to belong to the same right cell. There is a similar notion in the category of quantum tilting modules. The highest weights in tilting modules that generate the same tensor ideal are said to belong to the same weight cell.

DEFINITION 5.20. (Ostrik 2001) Write µ T_q λif Tq λ Q:Tq µ 0for some quantum tilting module Q.

REMARK5.21.Note that T_qis a preorder, since Tq λ Q1:Tq µ 0and Tq ν Q2:Tq λ 0gives Tq ν Q1 Q2:Tq µ 0. Note that Q1 Q2is tilting because a tensor product of tilting modules is tilting.

D^EFINITION 5.22. Let T_q be the equivalence relation defined by T_q. The equiv-alence classes of T_q are calledweight cells. The preorder T_q induces a partial order (also denoted T_q) on the set of weight cells in the natural way.

To ease the notation, we write ˇAfor the lower closure of an alcoveA.

REMARK5.23.

(i) If µ λ νand all three weights are dominant then µ T_qλas weight consid-erations show that Tq µ is a summand of Tq λ Tqν.

(ii) If Tq µ is a summand in a translation or a wallcrossing of Tq λ then µ T_qλ.

(iii) Let x W⁰and assume thatλ, µ xˇC X . Then (the proof of) Lemma 5.11 shows thatλ T_q µ. So a weight cell consists of all dominant weights in the lower closure of xC, x in some subset of W⁰.

(iv) If xy W⁰and y R x then y0 T_q x0since ΘsTq x0 :Tq y0 N_xH_s: N_y 0. This immediately give us that y R x implies y0 T_q x0and that y R x implies y0 T_qx0

THEOREM5.24. There is a one-to-one correspondence between the right cells in W⁰ and weight cells in X . The image of a right cell is given by

C ^"!$#

x C

xˇC X

This correspondence is order preserving.

P^ROOF. We first show that ^% x CxˇC X is a weight cell, for a right cellC_{. Letz,} y C_{and lety}λ yˇC X ,zµ zˇC X . We have

zµ T_qz0 T_qy0 T_qyλ

where the first and last T_q follows from Remark 5.23 (iii) and the second follows from (iv) asx R y.

To complete the proof we must show thatyλ yˇC,zµ zˇCwithzµ T_qyλgives z R y. First of all, we havez0 T_qy0 as before. This means thatTq z0 belongs to all tensor ideals containingTq y0; by Theorem 5.19 we conclude thatz R y. We get in the same wayy R zand hencez R y.

WEIGHT CELLS 43

Finally, it follows from Remark 5.23 (iv) that this mapping respects the order of the sets.

We will denote the weight cell corresponding to the ’first’ right cell e byc₁; note that c₁ C. The weight cell corresponding to the second right cellC_s0 is similarly denoted byc₂; it is the second largest weight cell. In the theorem below we identify the minimal weight cell which we will denote byc_St. Figure 5, 6, and 7 in Chapter 6 hold pictures of the weight cells in typeA2,B2, andG2.

THEOREM5.25. (Andersen 2001a)St X is a weight cell. It is minimal.

PROOF. The proof relies on a complete description of the injective indecomposable tilting modules inT_q, given in (Andersen 2001b). We have

Tqλ injective λ St X

Let us first see thatTqSt is in every tensor ideal of tilting modules. The dual of a tilting module is tilting, since the dual of Weyl module is an induced module and vice versa.

Further, the dual of a indecomposable module is indecomposable, as EndQ EndQ for all modules. We thus find thatTλ T w0λ, since the highest weight ofTλ is

w0λ. Then

HomTqStTqλ Tq w0λ St HomTqSt Tq w0λTq w0λ St The last space is nonzero since the second module is a summand in the first by weight considerations. ThenTqSt is a summand inTqλ Tq w0λ St asTqSt is simple and injective. It follows that St T_qλfor all dominant weights. So St is a minimal element in the preorder T_q.

We have seen that St belongs to the minimal weight cell. By Remark 5.23 (i) we find that St X is contained in this minimal weight cell. It remains to see thatλ T_qSt implies λ St X .

Suppose thatTqλ is a summand in aTqSt QwithQinT_q. The tensor product is injective since TqSt is. ThenTqλ is necessarily injective as it is a summand in an injective module. It follows thatλ St X .

CHAPTER 6

In document Afhandling (Sider 35-45)