A LECTURE ON FINITE WEYL GROUPOIDS, OBERWOLFACH 2012
|
|
- Noah Robinson
- 6 years ago
- Views:
Transcription
1 A LECTURE ON FINITE WEYL GROUPOIDS, OBERWOLFACH 202 M. CUNTZ Abstract. This is a short introduction to the theory of finite Weyl groupoids, crystallographic arrangements, and their classification. Introduction Reflections appear in many areas of mathematics. For instance, certain groups generated by involutions may be investigated by representing them as reflection groups. In particular, the Weyl groups belong to this class. They appear naturally inside semisimple algebraic groups and are fundamental for their classification. In their reflection representation, the Weyl groups are in fact subgroups of GL(Z r ) for some r. This integrality is a very strong and important restriction; reflection groups with this property are also called crystallographic. Closely related to the algebraic group is another important structure, the Lie algebra. Lie algebras arise in nature as vector spaces of linear transformations, for example differential operators. It turns out that finite dimensional semisimple complex Lie algebras decompose into a direct sum labeled by roots and a Cartan subalgebra. These roots are (up to signs) the normal vectors defining the reflection hyperplanes of a Weyl group. Again, we have an integrality property for the roots: Let A be the real hyperplane arrangement given by the orthogonal complements of the roots. Then this is a simplicial arrangement and for each chamber K, the roots labeling the walls of K form a simple system, and in particular all other roots are integer linear combinations of the roots in. So apparently the combinatorics of root systems and Weyl groups play an important role in mathematics and moreover, a certain integrality is an essential feature of these structures. In the last decades, the concept of a Lie algebra has been generalized in many directions. For example, deformations of Lie algebras called quantum groups have proved to be useful in physics. More generally the theory of Hopf algebras seems to be a further natural direction. Recent results on pointed Hopf algebras have led to yet another symmetry structure, the Weyl
2 2 M. CUNTZ groupoid. Again one has vectors called roots, but this time the object acting on the roots is in general a groupoid and not a group anymore. A remarkable fact is that even in this much more general setting, the above integrality still plays a crucial role. The Weyl groupoid historically appeared as an invariant needed to classify Nichols algebras. However, the situation has changed during the last year: Recent observations have considerably increased the importance of the Weyl groupoid. If the real roots of a Weyl groupoid form a finite root system (as defined in Section.3), then we will say that the Weyl groupoid is finite. As for root systems from Coxeter groups, a finite Weyl groupoid W defines a hyperplane arrangement: Fix an object a and its set of positive roots R a +. The arrangement associated to W and a is the set of orthogonal complements of the elements of R a + in R r. It turns out that these arrangements are simplicial, i.e. the complement of the union of these hyperplanes decomposes into open simplicial cones (see [HW]). Moreover, it turns out that in terms of simplicial arrangements the axioms of a finite Weyl groupoid reduce to one single integrality property (I) (see [Cun] or Section.2 for details). We call simplicial arrangements satisfying this axiom (I) crystallographic arrangements. Among other things, this explains why the class of arrangements obtained from Weyl groupoids is so large. In fact this class is so large that for example in rank three, 53 of the 67 known sporadic simplicial arrangements (see [Grü09]) over Q are crystallographic. Since the classification of simplicial arrangements in the real projective plane is still an open problem, it could be very valuable to have a complete classification of a large subclass. But also in higher rank generalizations of crystallographic arrangements could have a great impact. Like reflection arrangements, all crystallographic arrangements are free [BC2]. They could provide examples or counterexamples in geometry or topology, especially since they may also be viewed as compact smooth toric varieties (see [CRT2] or Section.6);. Weyl groupoids and crystallographic arrangements.. Simplicial arrangements. Let r N, V := R r. For α V we write α = ker(α). We first recall the definition of a simplicial arrangement (compare [OT92,.2, 5.]). Definition.. An arrangement of hyperplanes A is a finite set of hyperplanes in V. Let K(A) be the set of connected components (chambers) of V \ H A H. If every chamber K is an open simplicial cone,
3 A LECTURE ON FINITE WEYL GROUPOIDS 3 i.e. there exist α,..., αr V such that { r } K = a i αi a i > 0 for all i =,..., r =: α,..., αr >0, i= then A is called a simplicial arrangement. Example.2. () r = 2 and r = 3 (projective plane) (2) Let W be a real reflection group, R the set of roots of W. Then A = {α α R} is a simplicial arrangement..2. Crystallographic arrangements. Let A = {H,..., H n }, A = n be simplicial. For each H i, i =,..., n we choose an element x i V such that H i = x i and let R := {±x,..., ±x n } V. For each chamber K K(A) set B K = { normal vectors in R of the walls of K pointing to the inside }. If α,..., α r is the dual basis to B K = {α,..., α r }, then K = α,..., α r >0 since A is simplicial. We are now ready for the main definition. Definition.3. Let A be a simplicial arrangement and R V a finite set such that A = {α α R} and Rα R = {±α} for all α R. We call (A, R) a crystallographic arrangement if for all K K(A): R α B K Zα. Two crystallographic arrangements (A, R), (A, R ) in V are called equivalent if there exists ψ Aut(V ) with ψ(r) = R. We then write (A, R) = (A, R ). Example.4. () Let R be the set of roots of the root system of a crystallographic Coxeter group. Then ({α α R}, R) is a crystallographic arrangement. (2) If R + := {(, 0), (3, ), (2, ), (5, 3), (3, 2), (, ), (0, )}, then {α α R + } is a crystallographic arrangement.
4 4 M. CUNTZ Question.5. What are the symmetries of crystallographic arrangements?.3. Cartan schemes and Weyl groupoids. We now define the notion of a Weyl groupoid which was introduced by Heckenberger and Yamane [HY08] and reformulated in [CH09b]. Example.6. Let α = (, 0, 0), α 2 = (0,, 0), α 3 = (0, 0, ), R a + := {α, α 2, α 3, (0,, ), (0,, 2), (, 0, ), (,, ), (,, 2)}. c i,j := max{k kα i + α j R+}, a c i,i := 2 for i j, (c i,j ) i,j = This is a generalized Cartan matrix. It defines reflections via For instance, and σ i (α j ) = α j c ij α i for j =, 2, 3. σ = ( 0 ) σ (R a +) = { α, α 2, (, 0, ), (,, ), (2,, 2), α 3, (0,, ), (,, 2)}. The elements of σ (R a +) are positive or negative. Let R a = R a + R a + and R b = σ (R a ) =: R b + R b +. Again, one can construct a Cartan matrix from R b + and it gives new reflections. In this example, we obtain the diagram: A σ A σ A σ 0 σ A Definition.7. Let I := {,..., r} and {α i i I} the standard basis of Z I. A generalized Cartan matrix C = (c ij ) i,j I is a matrix in Z I I such that (M) c ii = 2 and c jk 0 for all i, j, k I with j k, (M2) if i, j I and c ij = 0, then c ji = 0.
5 A LECTURE ON FINITE WEYL GROUPOIDS 5 Definition.8. Let A be a non-empty set, ρ i : A A a map for all i I, and C a = (c a jk ) j,k I a generalized Cartan matrix in Z I I for all a A. The quadruple is called a Cartan scheme if C = C(I, A, (ρ i ) i I, (C a ) a A ) (C) ρ 2 i = id for all i I, (C2) c a ij = c ρ i(a) ij for all a A and i, j I. Definition.9. Let C = C(I, A, (ρ i ) i I, (C a ) a A ) be a Cartan scheme. For all i I and a A define σ a i Aut(Z I ) by (.) σ a i (α j ) = α j c a ijα i for all j I. The Weyl groupoid of C is the category W(C) such that Ob(W(C)) = A and the morphisms are compositions of maps σ a i with i I and a A, where σ a i is considered as an element in Hom(a, ρ i (a)). The cardinality of I is the rank of W(C). Definition.0. A Cartan scheme is called connected if its Weyl groupoid is connected, that is, if for all a, b A there exists w Hom(a, b). The Cartan scheme is called simply connected, if Hom(a, a) = {id a } for all a A. There is a straight forward notion of equivalence of Cartan schemes which we skip here. Let C be a Cartan scheme. For all a A let (R re ) a = {id a σ i σ ik (α j ) k N 0, i,..., i k, j I} Z I. The elements of the set (R re ) a are called real roots (at a). The pair (C, ((R re ) a ) a A ) is denoted by R re (C). A real root α (R re ) a, where a A, is called positive (resp. negative) if α N I 0 (resp. α N I 0). Definition.. Let C = C(I, A, (ρ i ) i I, (C a ) a A ) be a Cartan scheme. For all a A let R a Z I, and define m a i,j = R a (N 0 α i + N 0 α j ) for all i, j I and a A. We say that R = R(C, (R a ) a A ) is a root system of type C, if it satisfies the following axioms. (R) R a = R a + R a +, where R a + = R a N I 0, for all a A. (R2) R a Zα i = {α i, α i } for all i I, a A. (R3) σ a i (R a ) = R ρ i(a) for all i I, a A. (R4) If i, j I and a A such that i j and m a i,j is finite, then (ρ i ρ j ) ma i,j (a) = a.
6 6 M. CUNTZ The root system R is called finite if for all a A the set R a is finite. By [CH09b, Prop. 2.2], if R is a finite root system of type C, then R = R re, and hence R re is a root system of type C in that case. Remark.2. If C is a Cartan scheme and there exists a root system of type C, then C satisfies (C3) If a, b A and id Hom(a, b), then a = b..4. From simplicial arrangements to groupoids. Lemma.3. Let (A, R) be a crystallographic arrangement, K 0, K K(A) be adjacent chambers and let B K 0 = {α,..., α r }. Write B K = { α, β 2,..., β r } for some β 2,..., β r R. Then there exists a permutation τ S r with τ() = and such that for certain c 2,..., c r N 0. β i = c τ(i) α + α τ(i), Proof. Let σ be the linear map i = 2,..., r σ : V V, α α, α i β i for i = 2,..., r. With respect to the basis B K 0, σ is a matrix of the form c 2 c r 0. A 0 for some c 2,..., c r N 0 and A N (r ) (r ) 0. By the same argument with K 0 and K interchanged, with respect to { α, β 2,..., β r }, the α 2,..., α r have coefficients in N 0, thus A N (r ) (r ) 0, and we conclude that A is a permutation matrix. Definition.4. Let K, K be adjacent chambers and B K B K = {α}. Then by Lemma.3 there exist unique c β N 0, β B K \{α} such that ϕ K,K : B K B K, β c β α + β, α α is a bijection. Now fix a chamber K 0 and an ordering B K 0 = {α,..., α r }. Then for any sequence µ,..., µ m {,..., r} we get a unique chain of chambers K 0,..., K m such that K i and K i are adjacent and for i =,..., m. B K i B K i = {ϕ Ki,K i... ϕ K0,K (α µi )}
7 A LECTURE ON FINITE WEYL GROUPOIDS 7 For i = 0,..., m, denote by σ K i µ i+ the linear map given by for all α B K i. By Lemma.3 σ K i µ i+ (α) = ϕ Ki,K i+ (α) σ K i µ i+ Aut(V ), (σ K i µ i+ ) 2 = id. For a sequence µ,..., µ m we may abbreviate σ µm... σ µ2 σ K 0 µ := σ K m µ m... σ K µ 2 σ K 0 µ since K,..., K m are uniquely determined by the sequence of µ i. We now construct a Weyl groupoid for a given crystallographic arrangement (A, R) of rank r. Let I := {,..., r}. Fix a chamber K 0 K(A) and an ordering B K 0 = {α,..., α r }. Consider the set  := {(µ,..., µ m ) m N, µ,..., µ m I} and write a.ν for the sequence (µ,..., µ m, ν) if a = (µ,..., µ m ), ν I. We have a map π :  End(V ), (µ,..., µ m ) σ µm... σ µ2 σ K 0 µ which yields an equivalence relation on  via for v, w Â. Let A v w : π(v) = π(w) := Â/. Each sequence a = (µ,..., µ m )  determines a unique map ϕ a by ϕ a := ϕ Km,K m... ϕ K0,K where K 0,..., K m is the sequence of chambers corresponding to a. We write K a := K m. Definition.5. For each a  we construct a Matrix Ca Z r r in the following way: Let i, j I and let K be the chamber adjacent to K a with B Ka B K = {ϕ a (α i )}. By Lemma.3 there exist integers c i,,..., c i,r such that ϕ a.i (α j ) = c i,j ϕ a (α i ) + ϕ a (α j ). We set C a := (c i,j ) i,j r. Notice that C a = C b for a, b  with π(a) = π(b), so we obtain a unique Cartan matrix for each element of A, which we will also denote C a, a A.
8 8 M. CUNTZ We now define the maps ρ i : A A, i I. First, set ˆρ i :  Â, a a.i. Since π(a) = π(b) implies ϕ a = ϕ b for a, b A, this induces well-defined maps ρ i : A A, a a.i, where a is the equivalence class of a in A. Proposition.6. Let (A, R) be a crystallographic arrangement of rank r and K 0 a chamber. Then I, A, (ρ i ) i I, (C a ) a A defined as above define a Cartan scheme which we denote C = C(A, R, K 0 ). Further, we obtain a Weyl groupoid W(C) = W(A, R, K 0 ). We may now construct a root system of type C(A, R, K 0 ). Let a = (µ,..., µ m )  and let φ a : V R r be the coordinate map for elements of V with respect to the basis ϕ a (α ),..., ϕ a (α r ) (in this ordering). Set R a := φ a (R). Again, notice that R a = R b for a, b  with π(a) = π(b), so we get a unique set which we also denote R a for each element of a A. Proposition.7. The sets R a, a A as above are the real roots of C = C(A, R, K 0 ) and form a root system of type C..5. From finite Weyl groupoids to arrangements. Let C = C(I, A, (ρ i ) i I, (C a ) a A ) be a connected simply connected Cartan scheme for which the real roots are a finite root system R; we call W(C) a finite Weyl groupoid. Note that in this situation, A is finite by [CH09b, Lemma 2.]. For each a A we set A a := {α α R a }. Now fix some a A. Let B denote the standard basis of R I and B its dual basis. Then B R a and thus K 0 := B >0 K(A a ) because no hyperplane of A intersects K 0 by (R). Using the definition of σi a and (R) at ρ i (a) we see that σ a i (K 0 ) K(A a ), i I,
9 A LECTURE ON FINITE WEYL GROUPOIDS 9 as well. But σi a is a linear automorphism, so the chambers of A a and A ρi(a) are isomorphic via σi a. Now consider the set U := σ µm... σµ a (K 0 ). m N µ,...,µ m I By induction every subset σ µm... σµ a (K 0 ) is a chamber of A a, and all of them are open simplicial cones. By construction, U is connected and closed, and since A is finite, U = R I. Thus A a is a simplicial arrangement. Alternatively, one can obtain the simpliciality as a corollary to results about the poset (Hom(W(C), a), R ) where R is the weak order (see [HW]). Alltogether, we get: Theorem.8 (see [Cun]). Let A be the set of all crystallographic arrangements and C be the set of all connected simply connected Cartan schemes for which the real roots are a finite root system. Then the map A/ = C/ =, (A, R) C(A, R, K), where K is any chamber of A, is a bijection..6. Fans and toric varieties. Definition.9. Let N := Z r. A fan in N is a nonempty collection Σ of strongly convex rational polyhedral cones in N Z R satisfying the following conditions: () Every face of any σ Σ is contained in Σ. (2) For any σ, σ Σ, the intersection σ σ is a face of both σ and σ. Let M := Hom Z (N, Z), σ Σ, and σ := {x M Z R x, y 0 for all y σ}. Further, U σ := Spec(C[M σ ]). Theorem.20. For a fan Σ in N, we can naturally glue {U σ σ Σ} together to obtain a toric variety σ Σ U σ associated to (N, Σ). Theorem.2 ([CRT2]). The chambers of a crystallographic arrangement are the maximal cones of a fan. There is a one-to-one correspondence between the crystallographic arrangements and the strongly symmetric smooth toric varieties.
10 0 M. CUNTZ 2. Classification of finite Weyl groupoids Throughout this section, let C = C(I, A, (ρ i ) i I, (C a ) a A ) be a connected simply connected Cartan scheme for which the real roots are a finite root system R. The assumption simply connected is not a strong rectriction: It suffices to compute automorphism groups to determine all other such Cartan schemes which are not simply connected, see [CH09a], [CH2], [CH], and [CH0]. 2.. Weyl groupoids: Rank two. In rank two, the object change diagram of C is a cycle, 2? which is locally of the form ( ) ( ) σ 2 c σ 2 2 c3 c 2 2 c 2 2 ( ) σ 2 c3 c 4 2 σ 2 Hence a Cartan scheme of rank two is more or less given by a sequence (..., c, c 2, c 3,...), the characteristic sequence. The morphism σσ i 2 i+ is the linear map ( ) ( ) ( ) ( ) ( ) ( ) ci 0 ci = 0 c i c i+ ( ) ( ) ci ci+ = = η(c 0 0 i )η(c i+ ) for η(i) = ( ) i SL(2, Z). 0 Remark 2.. The columns of the matrices η(c i ) η(c i+k ) for k odd and appropriate i, are roots. In fact, one can prove: Proposition 2.2. A finite sequence (c,..., c n ) N n, n is a characteristic sequence if and only if η(c ) η(c n ) = id, the entries in the first column of η(c ) η(c i ) are nonnegative for all i < n. Proposition 2.3. Let n N and b,..., b n Z. If b i 2 for all i {,..., n}, then η(b ) η(b n ) does not have finite order.
11 A LECTURE ON FINITE WEYL GROUPOIDS Figure. A triangulation for the characteristic sequence (, 4, 2,, 3, 2, 2) Since η(0) 2 = id, we always have at least one entry in a characteristic sequence of length > 2. Using for all a, b we obtain η(a + )η()η(b + ) = η(a)η(b) Proposition 2.4 (see [CH09a]). The set N of all charateristic sequences may be constructed recursively in the following way: () (0, 0) N. (2) If (c,..., c n ) N, then (c 2, c 3,..., c n, c n, c ) N and (c n, c n,..., c 2, c ) N. (3) If (c,..., c n ) N, then (c +,, c 2 +, c 3,..., c n ) N. Now to the roots. Consider on N 2 0 the total ordering Q, where (a, a 2 ) Q (b, b 2 ) if and only if a b 2 a 2 b. A finite sequence (v,..., v n ) of vectors in N 2 0 is an F-sequence if and only if v < Q v 2 < Q < Q v n and one of the following holds. n = 2, v = (0, ), and v 2 = (, 0). n > 2 and there exists i {2, 3,..., n } such that v i = v i + v i+ and (v,..., v i.v i+,..., v n ) is an F-sequence. Proposition 2.5 (see [CH]). For a A the set R a + ordered by Q is an F-sequence Some more general facts. We now return to the case of arbitrary rank. Let a A. The last observation on root systems of rank two implies the following useful result: Theorem 2.6. Let α R a +. Then either α is simple, or it is the sum of two positive roots. The correspondence to crystallographic arrangements yields:
12 2 M. CUNTZ Lemma 2.7. Let α, β R a +. If α, β Q R a ±(N 0 α + N 0 β), then there exists b A and w Hom(a, b) such that w(α), w(β) are simple Weyl groupoids: Rank three More facts for rank three. View the roots as elements of Z 3 with the lexicographic ordering. Lemma 2.8. Let α, β R a + be minimal in α, β Q R a + with respect to. Then α, β Q R a ±(N 0 α + N 0 β). Theorem 2.9 (Weak Convexity). Let a A and α, β, γ R+. a If det(α, β, γ) 2 = and α, β are the two smallest elements of α, β R+, a then either α, β, γ are simple, or one of γ α, γ β is in R a. Roots on the plane (,, ) Example: α = (0,, 0), β = (0, 0, ), γ = (, 7, 3) Theorem 2.0 (Required roots). Let a A and α, β, γ R a +. Assume that det(α, β, γ) 2 = and that γ α, γ β / R a. Then γ + α R a or γ + β R a. Theorem 2. (Bound for the Cartan entries). All entries of the Cartan matrices are greater or equal to 7. Remark 2.2. In fact, all entries of the Cartan matrices are greater or equal to An algorithm to enumerate root systems of rank three. Function Enumerate(R) () If R defines a crystallographic arrangement, output R and continue. (2) Y := {α + β α, β R, α β}\r. (3) For all α Y with α > max R: (a) Compute all subgroupoids of rank two in R {α}. (b) If all Cartan entries are 7, all rank two root sets are F-sequences, and the Weak Convexity is satisfied, then call Enumerate(R {α}).
13 A LECTURE ON FINITE WEYL GROUPOIDS 3 Remark 2.3. We use Theorem Required roots as a further obstruction. The algorithm terminates and yields the result: Theorem 2.4 (see [CH2]). Up to equivalences, there are 55 irreducible crystallographic arrangements of rank three Weyl groupoids: Rank four to eight. With the knowledge about rank three, we can now enumerate crystallographic arrangements in higher ranks. Here is a rough sketch of the algorithm. Function Enumerate(R) () If R R is a root set, output R R and continue. (2) For all subspaces U generated by elements of R, check that R U could be extended to a root set. (3) Set Y := {α + β α, β R, α β}\r. (4) For all α Y with α > max R, call Enumerate(R α) Weyl groupoids: Higher rank. An analysis of Dynkin diagrams leads to a complete classification in rank > 8. Theorem 2.5 (see [CH0]). There are exactly three families of irreducible crystallographic arrangements: () The family of rank two parametrized by triangulations of a convex n-gons by non-intersecting diagonals. (2) For each rank r > 2, arrangements of type A r, B r, C r and D r, and a further series of r arrangements. (3) Further 74 sporadic arrangements of rank r, 3 r 8. References [BC2] Mohamed Barakat and Michael Cuntz, Coxeter and crystallographic arrangements are inductively free, Adv. Math. 229 (202), no., [CH09a] M. Cuntz and I. Heckenberger, Weyl groupoids of rank two and continued [CH09b] fractions, Algebra & Number Theory 3 (2009), ,, Weyl groupoids with at most three objects, J. Pure Appl. Algebra 23 (2009), no. 6, , 6, 8 [CH0], Finite Weyl groupoids, arxiv: v (200), 35 pp. 0, 3 [CH], Reflection groupoids of rank two and cluster algebras of type A, J. Combin. Theory Ser. A 8 (20), no. 4, ,
14 4 M. CUNTZ [CH2], Finite Weyl groupoids of rank three, Trans. Amer. Math. Soc. 364 (202), no. 3, , 3 [CRT2] M. Cuntz, Y. Ren, and G. Trautmann, Strongly symmetric smooth toric varieties, Kyoto J. Math. 52 (202), no. 3, , 9 [Cun] M. Cuntz, Crystallographic arrangements: Weyl groupoids and simplicial arrangements, Bull. London Math. Soc. 43 (20), no. 4, , 9 [Grü09] B. Grünbaum, A catalogue of simplicial arrangements in the real projective plane, Ars Math. Contemp. 2 (2009), no., [HW] István Heckenberger and Volkmar Welker, Geometric combinatorics of Weyl groupoids, J. Algebraic Combin. 34 (20), no., , 9 [HY08] I. Heckenberger and H. Yamane, A generalization of Coxeter groups, root systems, and Matsumoto s theorem, Math. Z. 259 (2008), no. 2, [OT92] P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300, Springer-Verlag, Berlin, Michael Cuntz, Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D Kaiserslautern, Germany address: cuntz@mathematik.uni-kl.de
arxiv: v2 [math.co] 11 Oct 2016
ON SUBSEQUENCES OF QUIDDITY CYCLES AND NICHOLS ALGEBRAS arxiv:1610.043v [math.co] 11 Oct 016 M. CUNTZ Abstract. We provide a tool to obtain local descriptions of quiddity cycles. As an application, we
More informationRoot systems. S. Viswanath
Root systems S. Viswanath 1. (05/07/011) 1.1. Root systems. Let V be a finite dimensional R-vector space. A reflection is a linear map s α,h on V satisfying s α,h (x) = x for all x H and s α,h (α) = α,
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification
More informationTORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS
TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS YUSUKE SUYAMA Abstract. We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationClassification of root systems
Classification of root systems September 8, 2017 1 Introduction These notes are an approximate outline of some of the material to be covered on Thursday, April 9; Tuesday, April 14; and Thursday, April
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationarxiv: v1 [math.co] 26 Mar 2018
ON THE TITS CONE OF A WEYL GROUPOID M. CUNTZ, B. MÜHLHERR, AND C. J. WEIGEL arxiv:1803.09521v1 [math.co] 26 Mar 2018 Abstract. We translate the axioms of a Weyl groupoid with (not necessarily finite) root
More informationEKT of Some Wonderful Compactifications
EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some
More informationLongest element of a finite Coxeter group
Longest element of a finite Coxeter group September 10, 2015 Here we draw together some well-known properties of the (unique) longest element w in a finite Coxeter group W, with reference to theorems and
More informationMATH 223A NOTES 2011 LIE ALGEBRAS 35
MATH 3A NOTES 011 LIE ALGEBRAS 35 9. Abstract root systems We now attempt to reconstruct the Lie algebra based only on the information given by the set of roots Φ which is embedded in Euclidean space E.
More informationClassification of semisimple Lie algebras
Chapter 6 Classification of semisimple Lie algebras When we studied sl 2 (C), we discovered that it is spanned by elements e, f and h fulfilling the relations: [e, h] = 2e, [ f, h] = 2 f and [e, f ] =
More informationEssays on the structure of reductive groups. Root systems. X (A) = Hom(A, G m ) t t t t i,
9:51 p.m. November 11, 2006 Essays on the structure of reductive groups Root systems Bill Casselman University of British Columbia cass@math.ubc.ca Suppose G to be a reductive group defined over an algebraically
More informationParameterizing orbits in flag varieties
Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.
More informationBRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP
Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number 1, June 2001, Pages 71 75 BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP HI-JOON CHAE Abstract. A Bruhat-Tits building
More informationEigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups
Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue
More informationLOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi
LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi Department of Mathematics, East China Normal University, Shanghai, 200062, China and Center for Combinatorics,
More informationarxiv: v5 [math.ra] 17 Sep 2013
UNIVERSAL GEOMETRIC CLUSTER ALGEBRAS NATHAN READING arxiv:1209.3987v5 [math.ra] 17 Sep 2013 Abstract. We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient
More informationIsomorphisms between pattern classes
Journal of Combinatorics olume 0, Number 0, 1 8, 0000 Isomorphisms between pattern classes M. H. Albert, M. D. Atkinson and Anders Claesson Isomorphisms φ : A B between pattern classes are considered.
More informationA complement to the theory of equivariant finiteness obstructions
F U N D A M E N T A MATHEMATICAE 151 (1996) A complement to the theory of equivariant finiteness obstructions by Paweł A n d r z e j e w s k i (Szczecin) Abstract. It is known ([1], [2]) that a construction
More informationA classification of sharp tridiagonal pairs. Tatsuro Ito, Kazumasa Nomura, Paul Terwilliger
Tatsuro Ito Kazumasa Nomura Paul Terwilliger Overview This talk concerns a linear algebraic object called a tridiagonal pair. We will describe its features such as the eigenvalues, dual eigenvalues, shape,
More informationarxiv: v1 [math.rt] 11 Sep 2009
FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationBackground on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described
More informationOn exceptional completions of symmetric varieties
Journal of Lie Theory Volume 16 (2006) 39 46 c 2006 Heldermann Verlag On exceptional completions of symmetric varieties Rocco Chirivì and Andrea Maffei Communicated by E. B. Vinberg Abstract. Let G be
More informationA SHORT COURSE IN LIE THEORY
A SHORT COURSE IN LIE THEORY JAY TAYLOR Abstract. These notes are designed to accompany a short series of talks given at the University of Aberdeen in the second half session of the academic year 2009/2010.
More informationON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY
ON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY YUFEI ZHAO Abstract. In this paper we discuss the Bruhat order of the symmetric group. We give two criteria for comparing elements in this
More informationLECTURE 3 Matroids and geometric lattices
LECTURE 3 Matroids and geometric lattices 3.1. Matroids A matroid is an abstraction of a set of vectors in a vector space (for us, the normals to the hyperplanes in an arrangement). Many basic facts about
More informationTwisted Projective Spaces and Linear Completions of some Partial Steiner Triple Systems
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 49 (2008), No. 2, 341-368. Twisted Projective Spaces and Linear Completions of some Partial Steiner Triple Systems Ma lgorzata
More informationALGEBRAIC PROPERTIES OF BIER SPHERES
LE MATEMATICHE Vol. LXVII (2012 Fasc. I, pp. 91 101 doi: 10.4418/2012.67.1.9 ALGEBRAIC PROPERTIES OF BIER SPHERES INGA HEUDTLASS - LUKAS KATTHÄN We give a classification of flag Bier spheres, as well as
More informationMAT 5330 Algebraic Geometry: Quiver Varieties
MAT 5330 Algebraic Geometry: Quiver Varieties Joel Lemay 1 Abstract Lie algebras have become of central importance in modern mathematics and some of the most important types of Lie algebras are Kac-Moody
More informationBLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)
BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that
More informationOn the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2
Journal of Lie Theory Volume 16 (2006) 57 65 c 2006 Heldermann Verlag On the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2 Hervé Sabourin and Rupert W.T. Yu Communicated
More informationThe rank of connection matrices and the dimension of graph algebras
The rank of connection matrices and the dimension of graph algebras László Lovász Microsoft Research One Microsoft Way Redmond, WA 98052 August 2004 Microsoft Research Technical Report TR-2004-82 Contents
More informationDemushkin s Theorem in Codimension One
Universität Konstanz Demushkin s Theorem in Codimension One Florian Berchtold Jürgen Hausen Konstanzer Schriften in Mathematik und Informatik Nr. 176, Juni 22 ISSN 143 3558 c Fachbereich Mathematik und
More informationarxiv: v1 [math.gr] 7 Jan 2019
The ranks of alternating string C-groups Mark Mixer arxiv:1901.01646v1 [math.gr] 7 Jan 019 January 8, 019 Abstract In this paper, string C-groups of all ranks 3 r < n are provided for each alternating
More informationA Characterization of (3+1)-Free Posets
Journal of Combinatorial Theory, Series A 93, 231241 (2001) doi:10.1006jcta.2000.3075, available online at http:www.idealibrary.com on A Characterization of (3+1)-Free Posets Mark Skandera Department of
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationClassification of discretely decomposable A q (λ) with respect to reductive symmetric pairs UNIVERSITY OF TOKYO
UTMS 2011 8 April 22, 2011 Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs by Toshiyuki Kobayashi and Yoshiki Oshima T UNIVERSITY OF TOKYO GRADUATE SCHOOL OF
More informationarxiv: v1 [quant-ph] 19 Jan 2010
A toric varieties approach to geometrical structure of multipartite states Hoshang Heydari Physics Department, Stockholm university 10691 Stockholm Sweden arxiv:1001.3245v1 [quant-ph] 19 Jan 2010 Email:
More informationQUIVERS AND LATTICES.
QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given
More informationA 2 G 2 A 1 A 1. (3) A double edge pointing from α i to α j if α i, α j are not perpendicular and α i 2 = 2 α j 2
46 MATH 223A NOTES 2011 LIE ALGEBRAS 11. Classification of semisimple Lie algebras I will explain how the Cartan matrix and Dynkin diagrams describe root systems. Then I will go through the classification
More informationEnhanced Dynkin diagrams and Weyl orbits
Enhanced Dynkin diagrams and Weyl orbits E. B. Dynkin and A. N. Minchenko February 3, 010 Abstract The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called
More informationIRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. Contents
IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS NEEL PATEL Abstract. The goal of this paper is to study the irreducible representations of semisimple Lie algebras. We will begin by considering two
More informationTopics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group
Topics in Representation Theory: SU(n), Weyl hambers and the Diagram of a Group 1 Another Example: G = SU(n) Last time we began analyzing how the maximal torus T of G acts on the adjoint representation,
More informationA short introduction to arrangements of hyperplanes
A short introduction to arrangements of hyperplanes survey Sergey Yuzvinsky University of Oregon Pisa, May 2010 Setup and notation By a hyperplane arrangement we understand the set A of several hyperplanes
More informationEIGENVECTORS FOR A RANDOM WALK ON A LEFT-REGULAR BAND
EIGENVECTORS FOR A RANDOM WALK ON A LEFT-REGULAR BAND FRANCO SALIOLA Abstract. We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups
More informationTHE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL LIE ALGEBRAS
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 67 (2006) 2006, Pages 225 259 S 0077-1554(06)00156-7 Article electronically published on December 27, 2006 THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL
More informationClimbing Eements in Finite Coxeter Groups
Dublin Institute of Technology ARROW@DIT Articles School of Mathematics 2010 Climbing Eements in Finite Coxeter Groups Colum Watt Dublin Institute of Technology, colum.watt@dit.ie Thomas Brady Dublin City
More informationMath 249B. Geometric Bruhat decomposition
Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique
More informationDissimilarity maps on trees and the representation theory of GL n (C)
Dissimilarity maps on trees and the representation theory of GL n (C) Christopher Manon Department of Mathematics University of California, Berkeley Berkeley, California, U.S.A. chris.manon@math.berkeley.edu
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationRank 4 toroidal hypertopes
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 15 (2018) 67 79 https://doi.org/10.26493/1855-3974.1319.375 (Also available at http://amc-journal.eu) Rank
More informationHomogeneous Coordinate Ring
Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in
More informationarxiv: v1 [math.rt] 15 Oct 2008
CLASSIFICATION OF FINITE-GROWTH GENERAL KAC-MOODY SUPERALGEBRAS arxiv:0810.2637v1 [math.rt] 15 Oct 2008 CRYSTAL HOYT AND VERA SERGANOVA Abstract. A contragredient Lie superalgebra is a superalgebra defined
More informationNew Bounds for Partial Spreads of H(2d 1, q 2 ) and Partial Ovoids of the Ree-Tits Octagon
New Bounds for Partial Spreads of H2d 1, q 2 ) and Partial Ovoids of the Ree-Tits Octagon Ferdinand Ihringer Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem,
More informationADE SURFACE SINGULARITIES, CHAMBERS AND TORIC VARIETIES. Meral Tosun
Séminaires & Congrès 10, 2005, p. 341 350 ADE SURFACE SINGULARITIES, CHAMBERS AND TORIC VARIETIES by Meral Tosun Abstract. We study the link between the positive divisors supported on the exceptional divisor
More informationSUMS OF UNITS IN SELF-INJECTIVE RINGS
SUMS OF UNITS IN SELF-INJECTIVE RINGS ANJANA KHURANA, DINESH KHURANA, AND PACE P. NIELSEN Abstract. We prove that if no field of order less than n + 2 is a homomorphic image of a right self-injective ring
More informationA method for construction of Lie group invariants
arxiv:1206.4395v1 [math.rt] 20 Jun 2012 A method for construction of Lie group invariants Yu. Palii Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia and Institute
More informationA PROOF OF BOREL-WEIL-BOTT THEOREM
A PROOF OF BOREL-WEIL-BOTT THEOREM MAN SHUN JOHN MA 1. Introduction In this short note, we prove the Borel-Weil-Bott theorem. Let g be a complex semisimple Lie algebra. One basic question in representation
More informationTridiagonal pairs in algebraic graph theory
University of Wisconsin-Madison Contents Part I: The subconstituent algebra of a graph The adjacency algebra The dual adjacency algebra The subconstituent algebra The notion of a dual adjacency matrix
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Updated: April 14, 017 Total Positivity The final and historically the original motivation is from the study of total positive matrices, which
More informationSELF-EQUIVALENCES OF DIHEDRAL SPHERES
SELF-EQUIVALENCES OF DIHEDRAL SPHERES DAVIDE L. FERRARIO Abstract. Let G be a finite group. The group of homotopy self-equivalences E G (X) of an orthogonal G-sphere X is related to the Burnside ring A(G)
More informationThe Terwilliger Algebras of Group Association Schemes
The Terwilliger Algebras of Group Association Schemes Eiichi Bannai Akihiro Munemasa The Terwilliger algebra of an association scheme was introduced by Paul Terwilliger [7] in order to study P-and Q-polynomial
More informationarxiv: v1 [math.rt] 14 Nov 2007
arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof
More information5 Quiver Representations
5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )
More informationSPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS
SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the
More informationLIE ALGEBRAS: LECTURE 7 11 May 2010
LIE ALGEBRAS: LECTURE 7 11 May 2010 CRYSTAL HOYT 1. sl n (F) is simple Let F be an algebraically closed field with characteristic zero. Let V be a finite dimensional vector space. Recall, that f End(V
More informationGroup Gradings on Finite Dimensional Lie Algebras
Algebra Colloquium 20 : 4 (2013) 573 578 Algebra Colloquium c 2013 AMSS CAS & SUZHOU UNIV Group Gradings on Finite Dimensional Lie Algebras Dušan Pagon Faculty of Natural Sciences and Mathematics, University
More informationCHRISTOS A. ATHANASIADIS, THOMAS BRADY, AND COLUM WATT
SHELLABILITY OF NONCROSSING PARTITION LATTICES CHRISTOS A. ATHANASIADIS, THOMAS BRADY, AND COLUM WATT Abstract. We give a case-free proof that the lattice of noncrossing partitions associated to any finite
More informationThe symmetric group action on rank-selected posets of injective words
The symmetric group action on rank-selected posets of injective words Christos A. Athanasiadis Department of Mathematics University of Athens Athens 15784, Hellas (Greece) caath@math.uoa.gr October 28,
More informationFORMALITY OF THE COMPLEMENTS OF SUBSPACE ARRANGEMENTS WITH GEOMETRIC LATTICES
FORMALITY OF THE COMPLEMENTS OF SUBSPACE ARRANGEMENTS WITH GEOMETRIC LATTICES EVA MARIA FEICHTNER AND SERGEY YUZVINSKY Abstract. We show that, for an arrangement of subspaces in a complex vector space
More informationarxiv: v1 [math.rt] 31 Oct 2008
Cartan Helgason theorem, Poisson transform, and Furstenberg Satake compactifications arxiv:0811.0029v1 [math.rt] 31 Oct 2008 Adam Korányi* Abstract The connections between the objects mentioned in the
More informationLECTURE 11: SOERGEL BIMODULES
LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing
More informationCategory O and its basic properties
Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α
More informationRoot system chip-firing
Root system chip-firing PhD Thesis Defense Sam Hopkins Massachusetts Institute of Technology April 27th, 2018 Includes joint work with Pavel Galashin, Thomas McConville, Alexander Postnikov, and James
More informationA quasisymmetric function generalization of the chromatic symmetric function
A quasisymmetric function generalization of the chromatic symmetric function Brandon Humpert University of Kansas Lawrence, KS bhumpert@math.ku.edu Submitted: May 5, 2010; Accepted: Feb 3, 2011; Published:
More informationREVERSALS ON SFT S. 1. Introduction and preliminaries
Trends in Mathematics Information Center for Mathematical Sciences Volume 7, Number 2, December, 2004, Pages 119 125 REVERSALS ON SFT S JUNGSEOB LEE Abstract. Reversals of topological dynamical systems
More informationSpin norm: combinatorics and representations
Spin norm: combinatorics and representations Chao-Ping Dong Institute of Mathematics Hunan University September 11, 2018 Chao-Ping Dong (HNU) Spin norm September 11, 2018 1 / 38 Overview This talk aims
More information58 CHAPTER 2. COMPUTATIONAL METHODS
58 CHAPTER 2. COMPUTATIONAL METHODS 23 Hom and Lim We will now develop more properties of the tensor product: its relationship to homomorphisms and to direct limits. The tensor product arose in our study
More informationTOPOLOGICALLY FREE ACTIONS AND PURELY INFINITE C -CROSSED PRODUCTS
Bull. Korean Math. Soc. 31 (1994), No. 2, pp. 167 172 TOPOLOGICALLY FREE ACTIONS AND PURELY INFINITE C -CROSSED PRODUCTS JA AJEONG 1. Introduction For a given C -dynamical system (A, G,α) with a G-simple
More informationUnless otherwise specified, V denotes an arbitrary finite-dimensional vector space.
MAT 90 // 0 points Exam Solutions Unless otherwise specified, V denotes an arbitrary finite-dimensional vector space..(0) Prove: a central arrangement A in V is essential if and only if the dual projective
More informationCanonical systems of basic invariants for unitary reflection groups
Canonical systems of basic invariants for unitary reflection groups Norihiro Nakashima, Hiroaki Terao and Shuhei Tsujie Abstract It has been known that there exists a canonical system for every finite
More informationWelsh s problem on the number of bases of matroids
Welsh s problem on the number of bases of matroids Edward S. T. Fan 1 and Tony W. H. Wong 2 1 Department of Mathematics, California Institute of Technology 2 Department of Mathematics, Kutztown University
More informationOn the Grassmann modules for the symplectic groups
On the Grassmann modules for the symplectic groups Bart De Bruyn Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 81 (S), B-9000 Gent, Belgium, E-mail: bdb@cage.ugent.be
More informationON COHERENCE OF GRAPH PRODUCTS AND COXETER GROUPS
ON COHERENCE OF GRAPH PRODUCTS AND COXETER GROUPS OLGA VARGHESE Abstract. Graph products and Coxeter groups are defined via vertex-edge-labeled graphs. We show that if the graph has a special shape, then
More informationRoot Systems Lie Algebras. Their Representations.
V Root Systems Lie Algebras and Their Representations. Kyoji Saito IPMU, the university of Tokyo August 4 and 5, 2009 Plan Lecture 1: Lecture 2: Lecture 3: Root Systems Lie algebras over a root system
More informationThe Symmetric Space for SL n (R)
The Symmetric Space for SL n (R) Rich Schwartz November 27, 2013 The purpose of these notes is to discuss the symmetric space X on which SL n (R) acts. Here, as usual, SL n (R) denotes the group of n n
More informationHILBERT BASIS OF THE LIPMAN SEMIGROUP
Available at: http://publications.ictp.it IC/2010/061 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL
More informationMath 249B. Nilpotence of connected solvable groups
Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationBruhat order, rationally smooth Schubert varieties, and hyperplane arrangements
FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, 833 840 Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements Suho Oh 1 and Hwanchul Yoo Department of Mathematics, Massachusetts
More informationRESEARCH ARTICLE. An extension of the polytope of doubly stochastic matrices
Linear and Multilinear Algebra Vol. 00, No. 00, Month 200x, 1 15 RESEARCH ARTICLE An extension of the polytope of doubly stochastic matrices Richard A. Brualdi a and Geir Dahl b a Department of Mathematics,
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
More informationA basis for the non-crossing partition lattice top homology
J Algebr Comb (2006) 23: 231 242 DOI 10.1007/s10801-006-7395-5 A basis for the non-crossing partition lattice top homology Eliana Zoque Received: July 31, 2003 / Revised: September 14, 2005 / Accepted:
More informationGroups of Prime Power Order with Derived Subgroup of Prime Order
Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*
More informationCLASSIFICATION OF TORIC MANIFOLDS OVER AN n-cube WITH ONE VERTEX CUT
CLASSIFICATION OF TORIC MANIFOLDS OVER AN n-cube WITH ONE VERTEX CUT SHO HASUI, HIDEYA KUWATA, MIKIYA MASUDA, AND SEONJEONG PARK Abstract We say that a complete nonsingular toric variety (called a toric
More informationA Study on Kac-Moody Superalgebras
ICGTMP, 2012 Chern Institute of Mathematics, Tianjin, China Aug 2o-26, 2012 The importance of being Lie Discrete groups describe discrete symmetries. Continues symmetries are described by so called Lie
More informationEuler characteristic of the truncated order complex of generalized noncrossing partitions
Euler characteristic of the truncated order complex of generalized noncrossing partitions D. Armstrong and C. Krattenthaler Department of Mathematics, University of Miami, Coral Gables, Florida 33146,
More information