Comptes Rendus Mathématique

José Matias

Pedro Miguel Santos

Elvira Zappale

Lower semicontinuity of nonlocal $L^\infty $ energies on $SBV_0(I)$

Aleksei B. Aleksandrov

Vladimir V. Peller

Comptes RendusMathématiqueAleksei B. Aleksandrov and Vladimir V. PellerRelatively bounded and relatively trace class perturbationsVolume 363 (2025), p. 377-382Online since: 20 May 2025https://doi.org/10.5802/crmath.722This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 377-382https://doi.org/10.5802/crmath.722Research article / Article de rechercheFunctional analysis, Harmonic analysis / Analyse fonctionnelle, AnalyseharmoniqueRelatively bounded and relatively trace classperturbationsPerturbations relativement bornées et relativement declasse traceAleksei B. Aleksandrov a,b and Vladimir V. Peller ,a,ba St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St. Petersburg, Russiab St. Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences,Fontanka 27, 191023 St. Petersburg, RussiaE-mails: aall54eexx@gmail.com, peller@math.msu.eduAbstract. In this note we study the behaviour of functions of self-adjoint operators under relatively boundedand relatively trace class perturbations. We introduce and study the class of relatively operator Lipschitzfunctions. We obtain a trace formula in the case of relatively trace class perturbations and show that thisclass of functions is the maximal class of functions for which the trace formula holds. Our method also givesus a new approach to the inequality∫ |ξ(t )|(1+ |t |)−1 dt < ∞ for the spectral shift function ξ in the case ofrelatively trace class perturbations.Résumé. Dans cette note nous étudions le comportement des fonctions d’opérateurs auto-adjoints sous desperturbations relativement dans la classe des opérateurs à trace et relativement bornées. Nous introduisonset étudions la classe de fonctions relativement lipschitziennes opératorielles. Nous obtenons une formule detrace dans le cas de perturbations relativement à trace et montrons que cette classe de fonctions est maximalepour la validité de la formule de trace. Notre méthode nous donne une nouvelle approche de l’inégalité∫ |ξ(t )|(1+|t |)−1 dt <∞ pour la fonction ξ de décalage spectral.Keywords. Relatively bounded perturbation, relatively trace class perturbation, relatively operator Lipschitzclass, trace formula, self-adjoint operators, double operator integrals.Mots-clés. Perturbations relativement bornées, perturbations relativement à trace, fonctions relativementlipschitziennes opératorielles, formules des traces, opérateurs auto-adjoints, intégrales d’opérateurs dou-bles.2020 Mathematics Subject Classification. 47A55.Funding. The research on Sections 2–4 is supported by the Russian Science Foundation (grant number 23-11-00153). The research on Sections 5–6 is supported by a grant of the Government of the Russian Federationfor the state support of scientific research, carried out under the supervision of leading scientists, agreement075-15-2024-631.Manuscript received 19 December 2024, revised 23 January 2025, accepted 24 January 2025.ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/378 Aleksei B. Aleksandrov and Vladimir V. Peller1. IntroductionThe purpose of this note is to study the behaviour of functions of self-adjoint operators underrelatively bounded and relatively trace class perturbations. It turns out that the natural classof functions to be considered in this connection is the class of relatively operator Lipschitzfunctions, which we are going to define in the introduction. We are going to obtain variouscharacterizations of this class. An important role will be played by double operator integrals.Let A be a not necessarily bounded self-adjoint operator in Hilbert space. A self-adjointoperator K (also, not necessarily bounded) is called a relatively bounded perturbation of A if∥K v∥ ≤ const(∥v∥+∥Av∥), v ∈ Dom(A), (1)where Dom(A) stands for the domain of the operator A. It is easy to see that property (1) isequivalent to the fact that the operatorCdef= K (A+ iI )−1 (2)is bounded. We say that a self-adjoint operator K is a relatively trace class perturbation of A if theoperator C defined by (2) belongs to the trace class S1.We consider parametric families At = A + tK , t ∈ R, where A and K are self-adjoint operatorsand K is a relatively bounded perturbation of A and we obtain a formula for the derivative of thefunction t 7→ f (At )− f (A) in the strong operator topology in terms of double operator integrals.This will be used to obtain a trace formula and compute the trace of f (A +K )− f (A) in thecase when K is a relatively trace class perturbation of A and f is a relatively operator Lipschitzfunction. Our results allow us to give a new approach to the inequality∫ |ξ(t )|(1+ |t |)−1 dt <∞,where ξ is the spectral shift function that corresponds to the pair {A, A+K }.Definition 1. A continuous function f on the real line R is called relatively operator Lipschitz ifthere is a positive number k such that∥∥ f (B)− f (A)∥∥≤ k∥∥(B − A)(A+ iI )−1∥∥ (3)whenever A and B are self-adjoint operators such that B − A is a relatively bounded perturbationof A.We denote by ROL the class of relatively operator Lipschitz functions on R.Lemma 2. Let f be a continuous function on R. It is relatively operator Lipschitz if and only iff (B)− f (A) ∈ S1whenever A and B are self-adjoint operators such that B − A is a relatively trace class perturbationof A.It is easy to verify that f is a relatively operator Lipschitz function if it is a resolvent Lipschitzfunction, i.e., ∥∥ f (B)− f (A)∥∥≤ const∥∥(B + iI )−1 − (A+ iI )−1∥∥,for arbitrary self-adjoint operators A and B .On the other hand, it is obvious that if f is a relatively operator Lipschitz function, then it is anoperator Lipschitz function, i.e., ∥∥ f (B)− f (A)∥∥≤ const∥A−B∥ (4)for arbitrary self-adjoint operators A and B , no matter bounded or unbounded. The class ofoperator Lipschitz functions on R is denoted by OL; the norm ∥ f ∥OL of a function f in OL is, bydefinition, the smallest possible constant on the right-hand side of (4).Aleksei B. Aleksandrov and Vladimir V. Peller 379We refer the reader to the survey [1] for detailed information about operator Lipschitz func-tions. It is well known (see [1]) that operator Lipschitz functions must be differentiable every-where on the real line R, and so relatively operator Lipschitz functions must also be differentiableon R.2. Characterizations of the class of relatively operator Lipschitz functionsIn this section we give several necessary and sufficient conditions for a function on R to berelatively operator Lipschitz. We also obtain similar characterizations of the class of resolventLipschitz functions. In Section 4 we obtain another characterization of the class of relativelyoperator Lipschitz functions in terms of Schur multipliers. Yet another characterization of theclass ROL (in terms of commutator estimates) will be given in Section 5.Definition 3. We say that a function f on R is a multiplier of the space OL of operator Lipschitzfunctions ifg ∈ OL =⇒ f g ∈ OL.We denote by MOL the class of all multipliers of OL.Theorem 4. Let f be a continuous function on R. The following are equivalent:(a) f ∈ ROL;(b) the function x 7→ (x + i) f (x) is an operator Lipschitz function;(c) f ∈MOL.The following theorem provides a similar characterization of the class of resolvent Lipschitzfunctions.Theorem 5. Let f be a continuous function on R. The following are equivalent:(a) f is a resolvent Lipschitz function;(b) the function x 7→ (x + i)2( f (x)− c) is an operator Lipschitz function for some c ∈C;(c) the function x 7→ (x + i)( f (x)− c) belongs to MOL for some c ∈C.If c ∈C satisfies (b) or (c), then c must be equal to lim|x|→∞ f (x).3. Double operator integralsAs in the case of operator Lipschitz functions (see [1]), an important role to study the class of rel-atively operator Lipschitz functions is played by double operator integrals. Double operator inte-grals appeared first in [7]; later Birman and Solomyak developed in [3–5] a rigorous mathematicaltheory of double operator integrals.Double operator integrals are expressions of the formÏΦ(x, y)dE1(x)Q dE2(y). (5)Here E1 and E2 are spectral measures on Hilbert space,Φ is a bounded measurable function andQ is a bounded linear operator on Hilbert space.The starting point of Birman and Solomyak in [3] was to define double operator integrals of theform (5) in the case when Q is a Hilbert–Schmidt operator, i.e., Q ∈ S2. In this caseÎΦdE1Q dE2also belongs to S2 and ∥∥∥∥Ï ΦdE1Q dE2∥∥∥∥S2≤ ∥Φ∥L∞∥Q∥S2 .380 Aleksei B. Aleksandrov and Vladimir V. PellerIfΦ has the propertyQ ∈ S1 =⇒ÏΦdE1Q dE2 ∈ S1,we say that Φ is a Schur multiplier with respect to the spectral measures E1 and E2. In this casewe can define by duality the double operator integral (5) for an arbitrary bounded linear operatorQ, see [1,3] for more detail.We use the notation M(R2) for the class of functions on R2 that are Schur multipliers withrespect to arbitrary Borel spectral measures.4. A characterization of the class of relatively operator Lipschitz functions in terms ofSchur multipliersRecall that for a differentiable function f on R, the divided difference D f is defined on R2 by(D f )(x, y)def=f (x)− f (y)x − y , if x ̸= y,f ′(x), if x = y.Put(D f )x(x, y)def= (x + i)(D f )(x, y), (x, y) ∈R2.The following result describes the class ROL in terms of Schur multipliers.Theorem 6. Let f be a continuous function on R. Then f is relatively operator Lipschitz if andonly if it is differentiable on R and (D f )x ∈M(R2).Note that the part “if” was established in [4, Theorem 4.6].Now we are in a position to obtain a useful formula for f (B)− f (A) in the case when B − A is arelatively bounded perturbation of A and f is a relatively operator Lipschitz function on R.Theorem 7. Let f be a relatively operator Lipschitz function on R. If A and B are self-adjointoperators such that B − A is a relatively bounded perturbation of A, thenf (B)− f (A) =ÏR×R(D f )(x, y)(y + i)dEB (x)(B − A)(A+ iI )−1 dE A(y).5. A characterization of the class of relatively operator Lipschitz functions in terms ofcommutatorsIn this section we are going to obtain a characterization of the class of relatively operatorLipschitz functions in terms of commutator and quasi-commutator estimates. The followingresult means, in particular, that a function is relatively operator Lipschitz if and only if it isrelatively commutator Lipschitz, i.e. satisfies statement (b) of the following theorem.Theorem 8. Let f : R→C be a differentiable function. The following statements are equivalent:(a) f is a relatively operator Lipschitz function;(b) ∥∥ f (A)R −R f (A)∥∥≤ const∥∥(AR −R A)(A+ iI )−1∥∥for any bounded operator R and any self-adjoint operator A;(c) ∥∥ f (B)R −R f (A)∥∥≤ const∥∥(BR −R A)(A+ iI )−1∥∥for any bounded operator R and any self-adjoint operators A and B.Aleksei B. Aleksandrov and Vladimir V. Peller 3816. Differentiation in a parameterTo obtain a trace formula for f (B)− f (A) in the case when K def= B − A is a self-adjoint relativelytrace class perturbation of A, we are going to consider a parametric family Atdef= A + tK , t ∈ R,and differentiate the functiont 7→ f (At )− f (A)in the parameter t in the strong operator topology.Theorem 9. Let f be a relatively operator Lipschitz function. Suppose that K is a relativelybounded perturbation of a self-adjoint operator A. Then the function t 7→ f (At )− f (A) is differen-tiable on R in the strong operator topology anddds(f (As )− f (At ))∣∣∣∣s=t=ÏR×Rf (x)− f (y)x − y (y + i)dE At (x)K (At + iI )−1 dE At (y) (6)for any t ∈R.7. The trace formulaFor a pair of self-adjoint operators {A,B} with trace class difference the very important Lifshits–Krein trace formula [8,9] gives the following expression for the trace of the difference f (A)− f (B):trace(f (A)− f (B))= ∫Rf ′(t )ξ(t )dtfor nice functions f , where ξ is the so-called spectral shift function for the pair {A,B}. It is real-valued and belongs to L1(R). It was established in [10] that the maximal class of functions f forwhich the Lifshits–Krein trace formula holds for arbitrary pairs of self-adjoint operators with traceclass difference coincides with the class of operator Lipschitz functions OL.To obtain a trace formula in the case of relatively operator Lipschitz functions and relativelytrace class perturbations, we are going to use an idea of Birman and Solomyak [5]. Namely, weare going to use formula (6) to compute the derivative of the function t 7→ f (At )− f (A) in thestrong operator topology. Then we compute the trace of the derivative and use a version of theNewton–Leibnitz formula to compute the trace of f (B)− f (A). We also describe the maximalclass of functions for which the trace formula holds in the case of arbitrary relatively trace classperturbations.Theorem 10. Let A be a self-adjoint operator and let K be a relatively trace class perturbation of A.Then there exists a unique measurable real-valued function ξ on R satisfying∫R|ξ(t )|1+|t | dt <∞ (7)and such that for an arbitrary relatively operator Lipschitz function f on R the following traceformula holdstrace(f (A+K )− f (A))= ∫Rf ′(t )ξ(t )dt . (8)The function ξ satisfying (8) and (7) is called the spectral shift function for the pair {A, A+K }.The fact that the spectral shift function in the case of a relatively trace class perturbation mustsatisfy (7) was established earlier in [6].Theorem 11. Let f be a differentiable function onR such that trace formula (8) holds for arbitraryself-adjoint operators A and K such that K is a relatively trace class perturbation of A. Thenf ∈ ROL.The details of this note can be found in [2].382 Aleksei B. Aleksandrov and Vladimir V. PellerDeclaration of interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.References[1] A. B. Aleksandrov and V. V. Peller, “Operator Lipschitz functions”, Russ. Math. Surv. 71(2016), no. 4, pp. 605–702.[2] A. B. Aleksandrov and V. V. Peller, “Functions of self-adjoint operators under relativelybounded and relatively trace class perturbations. Relatively operator Lipschitz functions”,2025. Online at https://arxiv.org/abs/2411.01901.[3] M. S. Birman and M. Z. Solomyak, “Double Stieltjes operator integrals”, Probl. Mat. Fiz. 1(1966), pp. 33–67.[4] M. S. Birman and M. Z. Solomyak, “Double Stieltjes operator integrals. III”, Probl. Mat. Fiz.6 (1973), pp. 27–53.[5] M. S. Birman and M. Z. Solomyak, “Remarks on the spectral shift function”, J. Sov. Math. 3(1975), pp. 408–419.[6] A. Chattopadhyay and A. Skripka, “Trace formulas for relative Schatten class perturba-tions”, J. Funct. Anal. 274 (2018), pp. 3377–3410.[7] Y. L. Daletskĭı and S. G. Kreı̆n, “Integration and differentiation of functions of Hermitianoperators and application to the theory of perturbations (Russian)”, Voronež. Gos. Univ.Trudy Sem. Funkcional. Anal. 1 (1956), pp. 81–105.[8] M. G. Kreı̆n, “On the trace formula in perturbation theory (Russian)”, Mat. Sb. 33 (1953),pp. 597–626.[9] I. M. Lifshits, “On a problem in perturbation theory connected with quantum statistics(Russian)”, Usp. Mat. Nauk 7 (1952), pp. 171–180.[10] V. V. Peller, “The Lifshits–Krein trace formula and operator Lipschitz functions”, Proc. Am.Math. Soc. 144 (2016), pp. 5207–5215.

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Comptes RendusMathématiquePaolo SentinelliReflections and quotients in Coxeter groupsVolume 363 (2025), p. 235-239Online since: 8 April 2025https://doi.org/10.5802/crmath.714This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 235-239https://doi.org/10.5802/crmath.714Research article / Article de rechercheAlgebra, Combinatorics / Algèbre, CombinatoireReflections and quotients in Coxeter groupsRéflexions et quotients dans les groupes de CoxeterPaolo Sentinelli aa Dipartimento di Matematica, Politecnico di Milano, Milan, ItalyE-mail: paolosentinelli@gmail.comAbstract. We present a formula relating the set of left descents of an element of a Coxeter group with the setsof left descents of its projections on maximal quotients indexed by simple right descents. This formula is aninstance of a general result involving quotients and descents.Résumé. Nous présentons une formule reliant l’ensemble des descentes gauches d’un élément d’un groupede Coxeter aux ensembles des descentes gauches de ses projections sur les quotients maximaux indexés parles descentes droites simples. Cette formule est un exemple d’un résultat général impliquant les quotients etles descentes.Manuscript received 15 May 2024, revised and accepted 19 December 2024.1. IntroductionA reflection t in a Coxeter system (W,S) is a conjugate of a generator, and t is a left descent ofan element w ∈ W if ℓ(t w) < ℓ(w), where ℓ is the length function. Denoting by TL(w) the set ofleft descents of w , it is well known that the right weak order ⩽R on W is isomorphic to the set{TL(w) : w ∈W}ordered by inclusion. In this note we prove the following formula:TL(v) = TL(u) ∪⋃s∈DR (u−1v)TL(vS\{s}),for all u, v ∈ W such that u ⩽R v . Here DR (w) is the set of right simple descent (right descentsof length one) of w , and w J is the element of minimal length in the coset wWJ , for any J ⊆ S.This formula is a special case of a general result we prove in Theorem 2. We then deduce somecorollaries; for example, our result implies that the set{wS\{s} : s ∈ DR (w)}admits a join and it isprecisely w (Corollary 5), i.e.w = ∨s∈DR (w)wS\{s},for all w ∈ W . We are regarding the poset (W,⩽R ) as a complete meet-semilattice, for which thejoins may not exist.We now fix notation and recall some definitions useful for the rest of the paper. We refer to thebook of Björner and Brenti [1] for Coxeter groups.Let (W,S) be a Coxeter system, i.e. a presentation of the group W given by a set S of involutivegenerators and relations encoded by a Coxeter matrix m : S ×S → {1,2, . . . ,∞}. A Coxeter matrixover S is a symmetric matrix which satisfies the following conditions for all s, t ∈ S:(1) m(s, t ) = 1 if and only if s = t ;(2) m(s, t ) ∈ {2,3, . . . ,∞} if s ̸= t .ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/236 Paolo SentinelliThe presentation (W,S) of the group W is then the following:{generators: S,relations: (st )m(s,t ) = e,for all s, t ∈ S, where e denotes the identity in W . The Coxeter matrix m attains the value ∞ at (s, t )to indicate that there is no relation between the generators s and t . The class of words expressingan element of W contains words of minimal length; the length function ℓ : W →N assigns to anelement w ∈ W such minimal length. The identity e is represented by the empty word and thenℓ(e) = 0. A reduced word or reduced expression for an element w ∈W is a word of minimal lengthrepresenting w . The set of reflections of (W,S) is defined by T := {w sw−1 : w ∈W, s ∈ S}. For J ⊆ Sand v ∈W , we letW J := {w ∈W : ℓ(w) < ℓ(w s) ∀ s ∈ J},DR (v) :={s ∈ S : ℓ(v s) < ℓ(v)}.With WJ we denote the subgroup of W generated by J ⊆ S; in particular, WS = W and W∅ = {e}.We let ⩽R and ⩽ be the right weak order and the Bruhat order on W , respectively. The coveringrelations of the right weak order are characterized as follows: u◁R v if and only if ℓ(u) < ℓ(v) andu−1v ∈ S. For w ∈W , defineTL(w) :={t ∈ T : ℓ(t w) < ℓ(w)},TR (w) :={t ∈ T : ℓ(w t ) < ℓ(w)}.The right weak order has the following property (see [1, Proposition 3.1.3]):u ⩽R v ⇐⇒ TL(u) ⊆ TL(v).The covering relations of the Bruhat order are characterized as follows: u◁v if and only ifℓ(u) = ℓ(v) − 1 and u−1v ∈ T . The posets (W,⩽R ) and (W,⩽) are graded with rank functionℓ and (W,⩽R ) ,→ (W,⩽). For J ⊆ S, each element w ∈ W factorizes uniquely as w = w J w J ,where w J ∈ W J , w J ∈ WJ and ℓ(w) = ℓ(w J )+ℓ(w J ); see [1, Proposition 2.4.4]. We consider theidempotent function P J : W →W defined byP J (w) = w J ,for all w ∈ W . This function is order-preserving for the Bruhat order (see [1, Proposition 2.5.1]).Moreover I ⊆ J ⊆ S implies P J ◦P I = P J (see [2, Lemma 5.4]). We letP (s) := P S\{s},for all s ∈ S.For the following proposition see [1, Corollary 2.6.2].Proposition 1. Let u, v ∈W . Thenu ⩽ v ⇐⇒ P (s)(u)⩽ P (s)(v) ∀ s ∈ DR (u).Now we recall some results about the standard geometric representation of a finitely generatedCoxeter group. Let (W,S) be a Coxeter system with |S| < ∞ and Coxeter matrix m. Let V be thefree R-vector space on the set{αs : s ∈ S}and σs ∈ End(V ) defined by σs (v) = v − 2(αs |v)αs ,where the bilinear form ( · | · ) is defined by setting (αs |αt ) := −cos(πm(s,t )), for all s, t ∈ S; thenthe assignment w 7→ σw := σs1 · · ·σsk , where s1 · · · sk is any reduced word of w ∈ W , provides afaithful representation W → GL(V ) (see, e.g. [1, Chapter 4]). Then we let w(v) := σw (v), for allv ∈ V , w ∈ W . Let Φ := {w(αs ) : w ∈ W, s ∈ S}, V + := {∑s∈S asαs : as ⩾ 0 ∀ s ∈ S} \ {0} and V − :={∑s∈S asαs : as ⩽ 0 ∀ s ∈ S}\{0}; then it is well known thatΦ=Φ+⊎Φ− = {αt : t ∈ T }⊎{−αt : t ∈ T },Φ+ ⊆V + and Φ− ⊆V −. It holds that t (v) = v −2(αt |v)αt and w(αt ) ∈Φ− if and only if w t < w , forall w ∈W , t ∈ T (see, e.g. [1, Section 4.4]). Moreover, if s ∈ S and t ∈ T \ {s}, then s(αt ) =αst s .Paolo Sentinelli 2372. Reflections and quotientsIn this section we prove the main result of this paper, i.e. the formula mentioned in the introduc-tion, as a corollary of the following theorem.Theorem 2. Let (W,S) be a Coxeter system and u, v ∈W such that u ⩽R v. If E ⊆P (S) is such that⋂J∈E J = S \ DR (u−1v), thenTL(v) = TL(u) ∪⋃J∈ETL(vJ ).Proof. It is known thatTL(x y) = TL(x)+xTL(y)x−1, (1)for all x, y ∈ W , where + is the symmetric difference. Hence, for any v ∈ W , we have that∅ = TL(v−1v) = TL(v−1)+ v−1TL(v)v ; this implies TR (v) = TL(v−1) = v−1TL(v)v , for all v ∈ W .Consider J ⊆ S. Then(v J )−1TL(v J )v J ∩WJ = TR (v J )∩WJ =∅. (2)Notice that, if ℓ(x y) = ℓ(x)+ℓ(y), the symmetric difference in (1) is a disjoint union. Thereforewe have thatTL(v) = TL(v J )⊎ v J TL(v J )(v J )−1, (3)for all v ∈W , J ⊆ S. Equalities (2) and (3) implyTL(vJ ) = TL(v) \ v J TL(v J )(v J )−1= TL(v) \ v J WJ (v J )−1= TL(v) \ vWJ v−1.(4)If u ⩽R v and w := u−1v , we have that TL(v) = TL(u)⊎uTL(w)u−1. Moreover, by (4),uTL(wJ )u−1 = u(TL(w) \ wWJ w−1)u−1= uTL(w)u−1 \ vWJ v−1⊆ TL(v) \ vWJ v−1 = TL(v J ).(5)Therefore, by (4) and (5), if we let K :=⋂J∈E J , we obtain⋃J∈ETL(vJ ) ⊇ ⋃J∈EuTL(wJ )u−1= u(⋃J∈E(TL(w) \ wWJ w−1))u−1= u(TL(w) \ wWK w−1)u−1= u(TL(w) \ wWS\DR (w)w−1)u−1= uTL(wS\DR (w))u−1= uTL(w)u−1.This implies thatTL(u) ∪⋃J∈ETL(vJ ) ⊇ TL(u)∪uTL(w)u−1 = TL(v).On the other hand, since u ⩽R v and v J ⩽R v for all J ∈ E , we also have that TL(v) ⊇ TL(u) ∪⋃J∈E TL(v J ) and this concludes the proof. □By the fact that⋂s∈DR (u−1v)(S \ {s}) = S \ DR (u−1v), we obtain the following corollary, which isthe finest instance of the statement of Theorem 2.238 Paolo SentinelliCorollary 3. Let (W,S) be a Coxeter system and u, v ∈W such that u ⩽R v. ThenTL(v) = TL(u) ∪⋃s∈DR (u−1v)TL(vS\{s}).In the following example we consider a Coxeter system of type B4. The set of generators isS = {s0, s1, s2, s3}; the Coxeter matrix takes the values m(s0, s1) = 4 and m(s1, s2) = m(s2, s3) = 3.Example 4. In type B4, let u = s2s3s2 and v = s2s3s2s1s0s2s3. Then u−1v = s1s0s2s3, DR (v) ={s0, s2, s3} and DR (u−1v) = {s0, s3}. We have that• TL(u) ={s2, s3, s2s3s2},• TL(vS\{s3})= TL(s1s2s3) = {s1, s1s2s1, s1s2s3s2s1},• TL(vS\{s0})= TL(s3s2s1s0) = {s3, s2s3s2, s1s2s3s2s1, s3s2s1s0s1s2s3},• TL(v) ={s1, s2, s3, s1s2s1, s2s3s2, s1s2s3s2s1, s3s2s1s0s1s2s3}.An element w ∈W is uniquely determined by its descent set; hence, by Theorem 2 for u = e, wehave that w is uniquely determined by the set{wS\{s} : s ∈ DR (w)}. This fact can be also deducedfrom Proposition 1. An additional information provided by Theorem 2 is the following one, whichconcerns the join operation in the right weak order.Corollary 5. Let w ∈W . Then w = ∨s∈DR (w)wS\{s}.Proof. Clearly TL(w J ) ⊆ TL(w), for all J ⊆ S, or, equivalently, w J ⩽R w , for all J ⊆ S. Hence theset{wS\{s} : s ∈ DR (w)}is bounded and then it admits a join w ′ (see [3, Theorem 1.5]). Moreover,by [4, Proposition 2.14] we have that TL(w ′) = TL(w) and this concludes the proof. □By repeated use of Corollary 3, we obtain the following result.Corollary 6. Let s1s2 · · · sk be a reduced expression for w ∈W . ThenTL(w) =k−1⋃i=0TL((s1 · · · sk−i )S\{sk−i }).We now deduce from Theorem 2 the statement of [5, Lemma 4.2]. Such result has inspired ourmain formula and it is related to the shellability of Coxeter complexes.Corollary 7. Let u, v, w ∈ W be such that u ⩽R v and u ⩽R w. If P (s)(v) = P (s)(w) for alls ∈ DR (u−1v), then v ⩽R w.Proof. Since u ⩽R w , we have that TL(u) ⊆ TL(w). Moreover TL(vS\{s}) = TL(wS\{s}) ⊆ TL(w)for all s ∈ DR (u−1v). Hence by Corollary 3 we obtain TL(v) ⊆ TL(w), which is equivalent tov ⩽R w . □For n ∈ N, let Bn be the power set P({1, . . . ,n})ordered by inclusion and, for w ∈ W andK ⊆ S \ DR (w),T Kw :={TL(wK∪J): J ⊆ DR (w)}⊆P (T ),ordered by inclusion.Proposition 8. Let w ∈W , K ⊆ S \ DR (w) and n := |DR (w)|. Then, as posets, T Kw ≃ Bn .Proof. If w = e the result is trivial. So let w ̸= e and H := {s ∈ S : s ⩽ w}. We consider thestandard geometric representation of the finitely generated Coxeter group WH . Then w ∈WH andassume TL(wK∪I)= TL(wK∪J ), for some I , J ⊆ DR (w). Consider the factorization w = wK∪J wK∪J .We have that s ∈ J ⊆ DR (w) if and only if s ∈ DR(wK∪J), i.e. if and only if wK∪J (αs ) ∈ Φ−. Letts := w sw−1; then αts =−w(αs ). We have that(wK∪J)−1(αts ) =−(wK∪J )−1w(αs ) =−wK∪J (αs ).Paolo Sentinelli 239Hences ∈ J ⇐⇒ s ∈ DR(wK∪J)⇐⇒ (wK∪J )−1(αts ) ∈Φ+⇐⇒ ts ̸∈ TL(wK∪J)⇐⇒ ts ̸∈ TL(wK∪I)⇐⇒ s ∈ DR(wK∪I)⇐⇒ s ∈ I .Therefore I = J . We have proved that |Tw | = |Bn | = 2n . This also proves that TL(wK∪I) ⊆TL(wK∪J)implies J ⊆ I . On the other hand, if I ⊆ J , then K ∪ I ⊆ K ∪ J and we have thatwK∪J = (P K∪J ◦P K∪I )(w) ⩽R wK∪I , i.e. TL(wK∪J )⊆ TL(wK∪I ). Then the poset Tw is isomorphicto the dual of Bn and the result follows since Bn is self-dual. □Proposition 8 implies that the atoms of the poset T S\DR (w)w are the sets TL(wS\{s}), withs ∈ DR (w). The proof of this proposition also shows that if s ∈ DR (w) thenw sw−1 ∈ TL(wS\{s})\⋃r∈DR (w)\{s}TL(wS\{r }),i.e. the union in Corollary 7 is minimal with respect to its indexing set.AcknowledgmentsI thank the anonymous referee for having generalized the original main result (now Corollary 3)and proposing a more elegant proof. I also thank Christophe Hohlweg for some useful conversa-tions and for having pointed out the result of Corollary 5.Declaration of interestsThe author does not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and has declared no affiliations other than their researchorganizations.References[1] A. Björner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics,Springer, 2005, pp. xiv+363.[2] V. V. Deodhar, “On some geometric aspects of Bruhat orderings. II. The parabolic analogueof Kazhdan–Lusztig polynomials”, J. Algebra 111 (1987), no. 2, pp. 483–506.[3] M. Dyer, “On the weak order of Coxeter groups”, Can. J. Math. 71 (2019), no. 2, pp. 299–336.[4] C. Hohlweg and J.-P. Labbé, “On inversion sets and the weak order in Coxeter groups”, Eur.J. Comb. 55 (2016), pp. 1–19.[5] P. Sentinelli, “Linearly shellable complexes”. Online at https://link.springer.com/article/10.1007/s11083-024-09688-4. To appear in Order.

Reflections and quotients in Coxeter groups

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Lower semicontinuity of nonlocal $L^\infty$ energies on $SBV_0(I)$