On the closures of orbits of fourth order matrix pencils
|
|
- Brianna Wilson
- 5 years ago
- Views:
Transcription
1 On the closures of orbits of fourth order matrix pencils Dmitri D. Pervouchine Abstract In this work we state a simple criterion for nilpotentness of a square n n matrix pencil with respect to the action of SL n (C) SL n (C) SL 2 (C). The orbits of matrix pencils are classified explicitly for n = 4 and the hierarchy of closures of nilpotent orbits is described. Also, we prove that the algebra of invariants of the action of SL n (C) SL n (C) SL 2 (C) on C n C n C 2 is isomorphic to the algebra of invariants of binary forms of degree n with respect to the action of SL 2 (C). 1. Introduction Throughout this paper we focus on the natural linear representation of the group G = SL n (C) SL n (C) SL 2 (C) in the complex vector space V = C n C n C 2. If bases in C n, C n, and C 2 are chosen, then the components T ijk of a tensor T V form two square matrices X and Y, whose entries are T ij1 and T ij2, respectively. An element of V can be regarded as a pair of complex n n matrices. Then it is called a matrix pencil of order n and is denoted by λx + µy, where λ and µ are varying coefficients. The polynomial morphism ϕ: V W = S n (C 2 ) that takes each matrix pencil λx + µy to the binary form det(λx +µy ), is dominant and G-equivariant if we assume that SL n (C) acts on W trivially. Hence, the corresponding morphism ϕ : C[W ] G C[V ] G of the algebras of invariants is an embedding. Two matrix pencils of order n are said to be equivalent (respectively, strictly equivalent) if they are equivalent with respect to the action of SL n (C) SL n (C) SL 2 (C) (respectively, the action of SL n (C) SL n (C)). Here we state some basic results on matrix pencils. The direct sum of a n 1 m 1 matrix pencil P 1 = λx 1 +µy 1 and a n 2 m 2 matrix pencil P 2 = λx 2 +µy 2 is the (n 1 +n 2 ) (m 1 +m 2 ) matrix pencil P 1 + P 2 = λ(x 1 X 2 ) + µ(y 1 Y 2 ), where Z 1 Z 2 denotes the diagonal block matrix composed of Z 1 and Z 2. A matrix pencil is said to be indecomposable if it cannot be represented as a direct sum of non-trivial matrix pencils. We also consider n 0 and 0 m matrix pencils; by this we mean that if such pencil is present in a direct sum, then the corresponding matrices are given rows or columns of zeroes [3]. A matrix pencil λx + µy is called regular if it is square and det(λx + µy ) is not equal to zero identically. Otherwise, the pencil is called singular. It is known that every matrix pencil is strictly equivalent to the pencil L ε L εq + L ε L ε q + R R s, (1) where ε 1,..., ε q and ε 1,..., ε q are the minimal indices of columns and rows, respectively (we recall their definitions in the next section); L r is an indecomposable singular r (r + 1) 1
2 matrix pencil......, (2) and R 1,..., R s are indecomposable regular pencils [3]. The set of indecomposable pencils in (1) is defined unambiguously up to a transposition. A matrix pencil is said to be completely singular if there are no regular terms in (1). Every matrix pencil P is strictly equivalent to the pencil P r + P s, where P r is a regular matrix pencil, and P s is a completely singular matrix pencil. We recall the definition of minimal indices of rows and columns. For any matrix pencil P = λx + µy we consider the C[λ, µ]-module Ker P that consists of all x(λ, µ) (C[λ, µ]) n such that (λx + µy )x(λ, µ) = 0 (3) for any λ, µ C. The module Ker P is free [3]. A fundamental set of solutions of (3) is a minimal system of homogeneous generators of Ker P. The degrees ε 1,..., ε k of (any) fundamental set of solutions of (3) sorted in ascending order are called the minimal indices of columns of λx + µy. The minimal indices of columns of P = λx + µy are called the minimal indices of rows of λx + µy. 2. Nilpotent matrix pencils From now on we discuss only square matrix pencils. Consider a linear representation of a reductive algebraic group H in a vector space M. An element x of M is called nilpotent if the closure O x of its orbit O x contains zero element. The set of all nilpotent elements of M is called the nullcone and is denoted by N M. The nullcone plays an important role in the theory of invariants: it has the greatest dimension among all fibers of the factorization map, and the modality of the action on any fiber does not exceed the modality of the action on the nullcone [1]. Lemma 1. A singular matrix pencil is nilpotent. Proof. First of all, if ε 1 = 0 or ε 1 = 0, then the canonical form of the pencil contains a row or a column of zeroes. Therefore, the pencil is nilpotent. Now we prove that if r = ε 1 > 0 and s = ε 1 > 0, then the closure of the pencil s orbit contains zero element. Multiply the columns of L r by t s, the columns of L s by t r 1, the rows of L r by t s, the rows of L s by t r+1 except for the last, and the last row of L s by t s. This transformation affects only the last element of the last row of L s : it is multiplied by t 1 r s. Taking the limit t we get a row of zeroes. This completes the proof. Theorem 1. The regular pencil λx +µy is nilpotent if and only if the binary form det(λx + µy ) is nilpotent. Proof. It is obvious that ϕ(n V ) N W. Now we prove that ϕ 1 (N W ) N V. Suppose that λx + µy is mapped to a nilpotent binary form. We need to show that λx + µy N V. 2
3 0 -q q Figure 1: The mechanical interpretation of (4). If λx + µy is singular, then λx + µy N V. Otherwise, λx + µy is equivalent to the pencil λe + µd, where E is the identity matrix and D is an upper triangular matrix with λ 1,..., λ n on the diagonal. Since O λe+µd O λe+µd, where D = diag(λ 1,..., λ n ), it is enough to prove that O λe+µd 0. It is well known that a binary from is nilpotent if and only if it contains a linear factor whose multiplicity is greater than the half of the degree of the form. Therefore, at least s = [n/2] + 1 of λ 1,..., λ n are equal to each other. We can assume that these are λ 1,..., λ s, that is, λ 1 =... = λ s. With a suitable transformation D D λ 1 E, they are equal to zero. Now we prove that λe + µd is nilpotent. Consider the one-parameter subgroup g t = (A t, B t, C t ), where A t = diag(t α 1,..., t α n ), B t = E, C t = diag(t q, t q ), and α i = 0. Then g t (λe + µd) = λ (diag(t q+α 1,..., t q+α n )) + µ (diag(0,..., λ s+1 t q+α s+1,..., λ n t q+α n )). In order to make g t (λe + µd) approach zero when t, we need to choose α 1,..., α n so that all powers of t in the previous formula are negative. Thus, the following conditions must hold: α 1,..., α n < q α s+1,..., α n < q. (4) α α n = 0 We can confirm that α 1,..., α n and q satisfying this condition exist using the ballance scale shown in figure 1. We shall distribute n material points of equal mass on it so that the scale stays in equilibrium. The first s points shall be placed to the left of the point q on the right arm, and the remaining n s points shall be placed to the left of the point q on the left arm. It is clear that the desired distribution exists if the right arm gets more points than the left one (s > n/2). 3. Classification of orbits In [2] we classified the nilpotent orbits of fourth order matrix pencils using embedding of the original representation in the adjoint representation of the simple Lie algebra of type E 7. In this section we develop a method that is applicable to arbitrary matrix pencils. Lemma 2. If the matrix pencil P = λx + µy is completely singular, then it is strictly equivalent to the pencil P = λ(t 1 X) + µ(t 2 Y ) for any t 1, t 2 C. Proof. Without loss of generality we can assume than P is L r + L s. It is enough to prove that λx + µy is strictly equivalent to λx + µ(ty ). We put γ = t r+s 2, A = diag(γ 1, γ 1 t 1,..., γ 1 t 1 r, γ, γt,..., γt s ) B = diag(γ, γt,..., γt r, γ 1, γ 1 t 1,..., γ 1 t 1 s ). (5) 3
4 It is clear that (A, B) SL n (C) SL n (C). Under the action of (A, B) the second matrix of λx + µy, that is, Y, is multiplied by t, and the first matrix doesn t change. Theorem 2. Singular matrix pencils P 1 and P 2 are equivalent if and only if P1 s is equivalent to P2 s and P 1 r is equivalent to kp 2 r for some k C. Proof. Let P 1 and P 2 be equivalent. It follows from [4, theorem 3] that P1 s is equivalent to P2 s with respect to the action of GL n(c) GL n (C), and P1 r is equivalent to P 2 r with respect to the action of GL n (C) GL n (C) GL 2 (C). Hence, P1 s is equivalent to P 2 s by lemma 2 and P1 r is equivalent to kp 2 r for some k C. Conversely, if P 1 r is equivalent to kp 2 r for some k C and P1 s is non-trivial, then P 1 is mapped to tp2 s + P 2 r with a suitable transformation that belongs to SL n (C). To conclude the proof, it remains to apply lemma 2. The classification of orbits of matrix pencils is now reduced to the classification of orbits of completely singular matrix pencils and the classification of orbits of regular matrix pencils. Since a completely singular matrix pencil is defined unambiguously by the minimal indices of its rows and columns, the former problem is reduced to combining the blocks L r and L s to fill a square matrix of a given size. The classification of orbits of regular matrix pencils can be obtained easily using the idea of elementary divisors [4]. However, we will need another method in order to describe the closures of the orbits. Every regular matrix pencil λx +µy is equivalent to the pencil λe +µd, where E is the identity matrix, and D is a Jordan matrix. Two matrix pencils λe + µd and λe + µd are equivalent with respect to the action of G if and only if there exist matrices A, B SL n (C), and C SL 2 (C) such that { A(c11 E + c 12 D)B 1 = E A(c 21 E + c 22 D)B 1 = D (6), where c ij are the entries of C 1. This condition is equivalent to the existence of the matrices A SL n (C) and C SL 2 (C) such that A(ψ C (D))A 1 = D and where ψ C is a fractional transformation defined by det(c 11 E + c 12 D) = 1, (7) ψ C (X) = (c 21 E + c 22 X)(c 11 E + c 12 X) 1. (8) Lemma 3. If matrix pencils λe + µd and λe + µd are equivalent and D is a Jordan matrix with eigenvalue α, then D also is a Jordan matrix, whose eigenvalue is ψ C (α). Proof. Let D be αe + N, where N is a nilpotent matrix. Then D = ψ C (D) = ( (c 21 + c 22 α)e + N )( (c 11 + c 12 α)e + N ) 1 = ψc (α)(e + γ 1 N + γ 2 N ). Since Ker N Ker N k, it follows that Ker (D αe) Ker (ψ C (D) ψ C (α)e). We have dim Ker (D αe) = dim Ker (ψ C (D) ψ C (α)e) = 1, since ψ C 1(ψ C (D)) = D. Therefore, D is a Jordan matrix, whose eigenvalue is ψ C (α). This lemma proves the following Theorem 3. The matrix pencils λe + µd and λe + µd are equivalent if and only if the eigenvalues of D can be mapped to the eigenvalues of D by a fractional transformation that satisfies (7) and the dimensions of the corresponding Jordan blocks are equal. 4
5 We conclude that the equivalence class of a matrix pencil is defined by 1) the set (α 1,..., α s ) of (distinct) eigenvalues defined up to a fractional transformation that satisfies (7) and the set (k 1,..., k s ) of their multiplicities 2) the partition (k i1,..., k ili ) of each of k i by the corresponding Jordan blocks. In the next section we will describe the hierarchy of closures of nilpotent orbits when n 4 (if n > 4, then the number of nilpotent orbits is infinite [5]). By theorem 1, a regular nilpotent matrix pencil of order n 4 has up to two distinct eigenvalues, and any pair of the eigenvalues can be mapped to another pair by a fractional transformation that satisfies (7). Therefore, the equivalence class of a regular nilpotent matrix pencil is defined by the partitions (k 1,..., k s ) and (k i1,..., k ili ) only. We denote it by R k1 (k 11,..., k 1l1 ) R ks (k s1,..., k sls ). The integers k ij that are equal to 0 or 1 are omitted in this expression. For instance, the matrix pencil that has Jordan blocks of order 1 and 2 with the eigenvalue α 1 and a Jordan block of order 1 with the eigenvalue α 2 α 1 is denoted by R 3 (2) + R 1. The classification of the orbits of the fourth order matrix pencils is shown in table 1. Here dim denotes the dimension of the orbit, R and r are the rank invariants (see section 4.), ε i and ε i are the minimal indices of rows and columns, respectively. In the canonical form of a matrix pencil the terms L 0 and L 0 are omitted, and P + P is abbreviated as 2P. For instance, 2L 1 stands for L 1 +L 1 +L 0 +L 0. The dimension of the orbit of a matrix pencil can be calculated if the dimension of its stationary subgroup is known. The later is equal to the dimension of the centralizer of the pencil in the tangent algebra of G. We calculated (using a computer) the rank of the linear system that defines the centralizer, thereby obtaining its dimension. 4. Closures of nilpotent orbits The hierarchy of the orbits closures is represented as a graph, whose vertices are orbits. The vertices O 1 and O 2 are connected with an arrow if and only if the closure of O 1 contains O 2 and there are no intermediate orbits, that is, orbits O such that O 1 O O 2. We obtained the graph for the fourth order matrix pencil using the following lemmas and the fact that the closure of an orbit can contain only lower-dimensional orbits. Lemma 4. If the closure of the orbit of P contains P, then dim Ker m P dim Ker m P, where Ker m Q denotes the vector space of the homogeneous elements of C[λ, µ]-module Ker Q of degree m. Proof. Let P t O P be a sequence of matrix pencils that converges to P. All dim Ker m P t are equal to each other, as P t belong to the same orbit. Since the Grassman manifold is compact, there exist a converging subsequence of Ker m P t, whose limit point belongs to Ker P. Therefore, dim Ker m P dim Ker m P. Corollary 1. If the orbit of P contains P and ε 1 (P ) = ε 1 ( P ),..., ε k 1 (P ) = ε k 1 ( P ), then ε k (P ) ε k ( P ). For a matrix pencil P = λx + µy we put r(p ) = min rk(λx + µy ) (λ,µ) (0,0) R(P ) = max rk(λx + µy ). (9) (λ,µ) 5
6 Canon. form dim R r ε i ε i Canon. form dim R r ε i ε i 1 R 1 + R 3 (3) L 1 + R R 4 (4) L 1 + R R 1 + R 3 (2) R 3 (3) R 4 (3) R 1 + R L 1 + L R 3 (2) L 1 + L L L L 1 + L L L L 1 + L 1 + R L L 2 + R L L 2 + R L 1 + R R 1 + L L 1 + R R 1 + L R R 4 (2, 2) R R R R 1 + R R 2 (2) L 1 + R 2 (2) R L 1 + R 2(2) L R 1 + R 2 (2) L R 4 (2) R Table 1: Nilpotent matrix pencils of order 4. It is clear that r(p ) and R(P ) are invariant under G. Lemma 5. If the orbit of P contains P, then r(p ) r( P ) and R(P ) R( P ). Proof. Let sequence P t = λx t + µy t O P converge to P = λ X + µȳ. It is obvious that r(p t ) = r(p ) and R(P t ) = R(P ). For every t we put M t = {(λ : µ) P 1 rk(λx t + µy t )) = r(p )}. The sequence M t of non-empty sets has at least one limit point (λ 0 : µ 0 ) in P 1. Then, we have r(p ) rk(λ 0 X + µ0 Ȳ ) r( P ). The inequality R(P ) R( P ) is trivial. Lemma 6. Let P = P r + Z m, where Z m is a zero matrix pencil of order m. If the closure of the orbit of P contains P = P r + Z m, then the number of distinct eigenvalues of P r is smaller than or equal to the number of distinct eigenvalues of P r. Proof. It follows from (6) that if the sequence g t = (A t, B t, C t ) of elements of G is such that lim g t P = P, then lim g tp r = P r, where A t and B t are main submatrices of A t and B t, and g t = (A t, B t, C t ). Therefore, the closure of SL 2 (C)-orbit of the binary form ϕ(p r ) contains ϕ( P r ). The number of distinct eigenvalues of P r is equal to the number of linear factors in ϕ(p r ), which does not increase when we take the limit. 6
7 R 4 (4) R 1 +R 3 (3) L 3 L 2 + L 1 T R 4 (3) R 1 +R 3 (2) L 2 + R 1 L 1 +L 1T + R 1 R 4 (2,2) L 1 + 2R 1 R 4 (2) R 1 +R 3 L 1 + R 2 (2) L 1 + R 2 3R 1 2L 1 R 1 + R 2 (2) L 2 L 1 +L 1 T R 3 (3) R 3 (2) R 1 + R 2 L 1 + R 1 R 4 2R 1 R 3 L 1 R 2 (2) R 1 R 2 0 Figure 2: The hierarchy of closures of nilpotent orbits. 7
8 We build the hierarchy of closures as follows. Running over all orbits sorted in ascending order by their dimensions, we check whether an orbit of smaller dimension belongs to the closure of the given orbit. In order to do this, we either construct the sequence that converges to the desired orbit or prove that it is impossible using lemmas 4-6. Here we give some examples: 1. L 2 / L 1 + L 1, since dim Ker 0 L 2 = 2 and dim Ker 0 (L 1 + L 1 ) = R 1 + R 3 / R 4 (4), since R 1 + R 3 has two distinct eigenvalues, while R 4 (4) has three. 3. 2L 1 / R 4 (2), since r(2l 1 ) > r(r 4 (2)). 4. R n (k) R n 1 (k 1) + R 1 and, in particular, R 4 (2) R 1 + R 3. t 2 λ λ λ + tµ λ λ when t 0. λ + tµ λ 5. L 2 2L 1. If ε 1 (P ) = ε 2 (P ) = 1, the fundamental set of solutions for P is {(µ, 0, λ, λ), (0, µ, µ, µ 1)}, ε 1 ( P ) = 2 and the fundamental set of solutions for P is {(µ 2 1, µ, λ, 0)}, then ( ) ( ) µ µ when t 0. tµ 0 The hierarchy of orbits closures is symmetric with respect to the operation of transposition of matrix pencil. It follows from theorem 3 that if the matrix pencil P is regular, then P is equivalent to P. Therefore, it is enough to apply transposition to the singular part of the pencil. In figure 2 we save some space by showing only a part of the graph. To get the complete graph one needs to transpose all pencils shown in figure 2, add a copy of the original graph to the graph obtained, and identify the vertices that have the same labels. New edges do not appear when two parts of the graph are merged. However, it doesn t imply that if P 1 and P 2 belong to the different parts of the graph, then O P1 is not contained in the closure of O P2 or vice versa. For instance, the closure of L 1 + 2R 1 contains L 1 + L 1, whose closure, in turn, contains both L 1 + R 1 and L 1 + R Invariants In this section we prove that the algebra of invariants of G acting in the space of square matrix pencils of order n is isomorphic to the algebra of invariants of SL 2 (C) acting in the space of the binary forms of degree n. Lemma 7. The extension ϕ : C[W ] G C[V ] G is integral. Proof. Denote by I the ideal of C[V ] G that is generated by the image under ϕ of the maximal homogeneous ideal of C[W ] G. It follows from theorem 1 and Hilbert s Nullstellensatz that the radical of I coincides with the ideal of C[V ] G that is generated by all its homogeneous elements of positive degrees. Then by Hilbert s theorem [6] the extension ϕ : C[W ] G C[V ] G is integer. 8
9 Theorem 4. The extension ϕ : C[W ] G C[V ] G is an isomorphism. Proof. It follows from theorem 3 that a regular generic matrix pencil can be transformed to the pencil λe + µd, where E is the identity matrix, and D is diagonal matrix whose eigenvalues are distinct. The matrix pencil λe + µd is equivalent to the matrix pencil λe+µd if and only if there exist a fractional transformation that satisfies (7) and transforms the diagonal elements of D to the diagonal elements of D. The binary forms (λ + µx i ) and (λ + µy i ) are equivalent if and only if {x 1,..., x n } and {y 1,..., y n } are obtained from each other by a fractional transformation that preserves the highest coefficient of the form. It is clear that the later condition is equivalent to (7). Therefore, ϕ establishes oneto-one correspondence between generic orbits of G in V and W. Hence, it induces the isomorphism of the fields of rational G-invariants on V and W. These fields coincide with the corresponding quotient fields, as C[W ] and C[V ] are factorial and G doesn t have nontrivial characters [1, theorem 3.3]. To conclude the proof, it now remains to apply lemma 7 and note that C[W ] G is integrally closed. The author acknowledges Steven Epstein for carefully reading the translated manuscript. References [1] E. B. Vinberg and V. L. Popov, Invariant theory, Itogi Nauki i Tekhniki. Sovrem. problemy Matematiki, Fundamental nye napravleniya, vol. 55, VINITI, Moscow 1989, pp [2] D. D. Pervushin, Invariants and orbits of the standard (SL 4 (C) SL 4 (C) SL 2 (C))- module, Izv. Ross. Akad. Nauk Ser. Mat. 64:5 (2000), ; English transl., Izvestiya Math. 64:5, [3] Gantmacher F. R., The Theory of Matrices, Chelsea, New York, [4] Ja ja J., An addendum to Kronecker s theory of pencils, SIAM J. Appl. Math. 37:3 (1979), [5] Kac V. G., Some remarks on nilpotent orbits, Journal of Algebra 64 (1980), [6] D. Hilbert, Über die vollen Invarianten Systeme, Math. Ann. 42 (1893)
16.2. Definition. Let N be the set of all nilpotent elements in g. Define N
74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the
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 informationLECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)
LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is
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 informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
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 informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationRESEARCH STATEMENT FEI XU
RESEARCH STATEMENT FEI XU. Introduction Recall that a variety is said to be rational if it is birational to P n. For many years, mathematicians have worked on the rationality of smooth complete intersections,
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 information1. Introduction. Throughout this work we consider n n matrix polynomials with degree k of the form
LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES FERNANDO DE TERÁN, FROILÁN M. DOPICO, AND D. STEVEN MACKEY Abstract. A standard way of dealing with a regular matrix polynomial
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationCommuting nilpotent matrices and pairs of partitions
Commuting nilpotent matrices and pairs of partitions Roberta Basili Algebraic Combinatorics Meets Inverse Systems Montréal, January 19-21, 2007 We will explain some results on commuting n n matrices and
More informationThe number of simple modules of a cellular algebra
Science in China Ser. A Mathematics 2005 Vol.?? No. X: XXX XXX 1 The number of simple modules of a cellular algebra LI Weixia ( ) & XI Changchang ( ) School of Mathematical Sciences, Beijing Normal University,
More informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationSymplectic varieties and Poisson deformations
Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the
More informationMathematics 7800 Quantum Kitchen Sink Spring 2002
Mathematics 7800 Quantum Kitchen Sink Spring 2002 4. Quotients via GIT. Most interesting moduli spaces arise as quotients of schemes by group actions. We will first analyze such quotients with geometric
More informationREFLEXIVITY OF THE SPACE OF MODULE HOMOMORPHISMS
REFLEXIVITY OF THE SPACE OF MODULE HOMOMORPHISMS JANKO BRAČIČ Abstract. Let B be a unital Banach algebra and X, Y be left Banach B-modules. We give a sufficient condition for reflexivity of the space of
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationREPRESENTATION THEORY. WEEKS 10 11
REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.
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 informationSYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS
1 SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS HUAJUN HUANG AND HONGYU HE Abstract. Let G be the group preserving a nondegenerate sesquilinear form B on a vector space V, and H a symmetric subgroup
More informationREPRESENTATIONS OF S n AND GL(n, C)
REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although
More informationDefinition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).
9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationLECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)
LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups
More informationOn the Dimension of the Stability Group for a Levi Non-Degenerate Hypersurface
1 On the Dimension of the Stability Group for a Levi Non-Degenerate Hypersurface Vladimir Ezhov and Alexander Isaev We classify locally defined non-spherical real-analytic hypersurfaces in complex space
More informationALGEBRAIC GROUPS J. WARNER
ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic
More informationTHE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES
Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationGEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS
GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action
More informationLinear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More informationMargulis Superrigidity I & II
Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and Ruhr-Universität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationMath 396. Quotient spaces
Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationThe Hilbert-Mumford Criterion
The Hilbert-Mumford Criterion Klaus Pommerening Johannes-Gutenberg-Universität Mainz, Germany January 1987 Last change: April 4, 2017 The notions of stability and related notions apply for actions of algebraic
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationMath 110, Summer 2012: Practice Exam 1 SOLUTIONS
Math, Summer 22: Practice Exam SOLUTIONS Choose 3/5 of the following problems Make sure to justify all steps in your solutions Let V be a K-vector space, for some number field K Let U V be a nonempty subset
More informationHighest-weight Theory: Verma Modules
Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 430 (2009) 579 586 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Low rank perturbation
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 informationLIE ALGEBRAS: LECTURE 3 6 April 2010
LIE ALGEBRAS: LECTURE 3 6 April 2010 CRYSTAL HOYT 1. Simple 3-dimensional Lie algebras Suppose L is a simple 3-dimensional Lie algebra over k, where k is algebraically closed. Then [L, L] = L, since otherwise
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 informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationIrreducible subgroups of algebraic groups
Irreducible subgroups of algebraic groups Martin W. Liebeck Department of Mathematics Imperial College London SW7 2BZ England Donna M. Testerman Department of Mathematics University of Lausanne Switzerland
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationMultiplicity-Free Products of Schur Functions
Annals of Combinatorics 5 (2001) 113-121 0218-0006/01/020113-9$1.50+0.20/0 c Birkhäuser Verlag, Basel, 2001 Annals of Combinatorics Multiplicity-Free Products of Schur Functions John R. Stembridge Department
More informationOrthogonality graph of the algebra of upper triangular matrices *
Orthogonality graph of the algebra of upper triangular matrices * arxiv:602.03236v [math.co] 0 Feb 206 B.R. Bakhadly Lomonosov Moscow State University, Faculty of Mechanics and Mathematics Abstract We
More informationarxiv: v4 [math.rt] 9 Jun 2017
ON TANGENT CONES OF SCHUBERT VARIETIES D FUCHS, A KIRILLOV, S MORIER-GENOUD, V OVSIENKO arxiv:1667846v4 [mathrt] 9 Jun 217 Abstract We consider tangent cones of Schubert varieties in the complete flag
More information4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;
4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length
More informationAN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE
AN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE HENNING KRAUSE Abstract. Given an abelian length category A, the Gabriel-Roiter measure with respect to a length function l is characterized
More informationMath 215B: Solutions 1
Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an
More informationLattices and Hermite normal form
Integer Points in Polyhedra Lattices and Hermite normal form Gennady Shmonin February 17, 2009 1 Lattices Let B = { } b 1,b 2,...,b k be a set of linearly independent vectors in n-dimensional Euclidean
More informationRepresentations of Matrix Lie Algebras
Representations of Matrix Lie Algebras Alex Turzillo REU Apprentice Program, University of Chicago aturzillo@uchicago.edu August 00 Abstract Building upon the concepts of the matrix Lie group and the matrix
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationA short proof of Klyachko s theorem about rational algebraic tori
A short proof of Klyachko s theorem about rational algebraic tori Mathieu Florence Abstract In this paper, we give another proof of a theorem by Klyachko ([?]), which asserts that Zariski s conjecture
More informationMathematische Annalen
Math. Ann. (2012) 353:1273 1281 DOI 10.1007/s00208-011-0715-7 Mathematische Annalen On the rationality of the moduli space of Lüroth quartics Christian Böhning Hans-Christian Graf von Bothmer Received:
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationLECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 18. Category O for rational Cherednik algebra We begin to study the representation theory of Rational Chrednik algebras. Perhaps, the class of representations
More informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationLIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS
LIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS DIETRICH BURDE Abstract. We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify
More informationGeometry of Schubert Varieties RepNet Workshop
Geometry of Schubert Varieties RepNet Workshop Chris Spencer Ulrich Thiel University of Edinburgh University of Kaiserslautern 24 May 2010 Flag Varieties Throughout, let k be a fixed algebraically closed
More information0.1 Rational Canonical Forms
We have already seen that it is useful and simpler to study linear systems using matrices. But matrices are themselves cumbersome, as they are stuffed with many entries, and it turns out that it s best
More informationYOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP
YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP YUFEI ZHAO ABSTRACT We explore an intimate connection between Young tableaux and representations of the symmetric group We describe the construction
More informationPascal Eigenspaces and Invariant Sequences of the First or Second Kind
Pascal Eigenspaces and Invariant Sequences of the First or Second Kind I-Pyo Kim a,, Michael J Tsatsomeros b a Department of Mathematics Education, Daegu University, Gyeongbu, 38453, Republic of Korea
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationCOMMUTATIVE/NONCOMMUTATIVE RANK OF LINEAR MATRICES AND SUBSPACES OF MATRICES OF LOW RANK
Séminaire Lotharingien de Combinatoire 52 (2004), Article B52f COMMUTATIVE/NONCOMMUTATIVE RANK OF LINEAR MATRICES AND SUBSPACES OF MATRICES OF LOW RANK MARC FORTIN AND CHRISTOPHE REUTENAUER Dédié à notre
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 informationConvexity of the Joint Numerical Range
Convexity of the Joint Numerical Range Chi-Kwong Li and Yiu-Tung Poon October 26, 2004 Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement. Abstract Let A = (A 1,..., A m ) be an
More informationLinear and Bilinear Algebra (2WF04) Jan Draisma
Linear and Bilinear Algebra (2WF04) Jan Draisma CHAPTER 3 The minimal polynomial and nilpotent maps 3.1. Minimal polynomial Throughout this chapter, V is a finite-dimensional vector space of dimension
More informationMAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems.
MAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems. Problem 1 Find all homomorphisms a) Z 6 Z 6 ; b) Z 6 Z 18 ; c) Z 18 Z 6 ; d) Z 12 Z 15 ; e) Z 6 Z 25 Proof. a)ψ(1)
More informationPART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS
PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS Contents 1. Regular elements in semisimple Lie algebras 1 2. The flag variety and the Bruhat decomposition 3 3. The Grothendieck-Springer resolution 6 4. The
More informationThroughout these notes we assume V, W are finite dimensional inner product spaces over C.
Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal
More information1.4 Solvable Lie algebras
1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)
More information(f + g)(s) = f(s) + g(s) for f, g V, s S (cf)(s) = cf(s) for c F, f V, s S
1 Vector spaces 1.1 Definition (Vector space) Let V be a set with a binary operation +, F a field, and (c, v) cv be a mapping from F V into V. Then V is called a vector space over F (or a linear space
More informationSheaf cohomology and non-normal varieties
Sheaf cohomology and non-normal varieties Steven Sam Massachusetts Institute of Technology December 11, 2011 1/14 Kempf collapsing We re interested in the following situation (over a field K): V is a vector
More informationALGEBRAIC GEOMETRY I - FINAL PROJECT
ALGEBRAIC GEOMETRY I - FINAL PROJECT ADAM KAYE Abstract This paper begins with a description of the Schubert varieties of a Grassmannian variety Gr(k, n) over C Following the technique of Ryan [3] for
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology
More informationLUCK S THEOREM ALEX WRIGHT
LUCK S THEOREM ALEX WRIGHT Warning: These are the authors personal notes for a talk in a learning seminar (October 2015). There may be incorrect or misleading statements. Corrections welcome. 1. Convergence
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationHARTSHORNE EXERCISES
HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationLECTURE 16: REPRESENTATIONS OF QUIVERS
LECTURE 6: REPRESENTATIONS OF QUIVERS IVAN LOSEV Introduction Now we proceed to study representations of quivers. We start by recalling some basic definitions and constructions such as the path algebra
More informationA SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES
A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES PATRICK BROSNAN Abstract. I generalize the standard notion of the composition g f of correspondences f : X Y and g : Y Z to the case that X
More informationALGEBRA QUALIFYING EXAM PROBLEMS
ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General
More informationH(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).
92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported
More informationarxiv: v1 [math.rt] 7 Oct 2014
A direct approach to the rational normal form arxiv:1410.1683v1 [math.rt] 7 Oct 2014 Klaus Bongartz 8. Oktober 2014 In courses on linear algebra the rational normal form of a matrix is usually derived
More informationAlgebras and regular subgroups. Marco Antonio Pellegrini. Ischia Group Theory Joint work with Chiara Tamburini
Joint work with Chiara Tamburini Università Cattolica del Sacro Cuore Ischia Group Theory 2016 Affine group and Regular subgroups Let F be any eld. We identify the ane group AGL n (F) with the subgroup
More informationSYMMETRIES OF SECTIONAL CURVATURE ON 3 MANIFOLDS. May 27, 1992
SYMMETRIES OF SECTIONAL CURVATURE ON 3 MANIFOLDS Luis A. Cordero 1 Phillip E. Parker,3 Dept. Xeometría e Topoloxía Facultade de Matemáticas Universidade de Santiago 15706 Santiago de Compostela Spain cordero@zmat.usc.es
More informationSemistable Representations of Quivers
Semistable Representations of Quivers Ana Bălibanu Let Q be a finite quiver with no oriented cycles, I its set of vertices, k an algebraically closed field, and Mod k Q the category of finite-dimensional
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 informationMAIN ARTICLES. In present paper we consider the Neumann problem for the operator equation
Volume 14, 2010 1 MAIN ARTICLES THE NEUMANN PROBLEM FOR A DEGENERATE DIFFERENTIAL OPERATOR EQUATION Liparit Tepoyan Yerevan State University, Faculty of mathematics and mechanics Abstract. We consider
More informationOn conjugacy classes of the Klein simple group in Cremona group
Loughborough University Institutional Repository On conjugacy classes of the Klein simple group in Cremona group This item was submitted to Loughborough University's Institutional Repository by the/an
More information