An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula
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The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Bijective Proofs for Schur Function Identities Which Imply Dodgson's Condensation Formula and Plücker Relations
Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Pl¨ucker relations Markus Fulmek Institut f¨urMathematik der Universit¨atWien Strudlhofgasse 4, A-1090 Wien, Austria [email protected] Michael Kleber Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] Submitted: July 3, 2000; Accepted: March 7, 2001. MR Subject Classifications: 05E05 05E15 Abstract We present a “method” for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi–Trudi identity. We illustrate this “method” by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson’s condensation formula, Pl¨ucker relations and a recent identity of the second author. 1 Introduction Usually, bijective proofs of determinant identities involve the following steps (cf., e.g, [19, Chapter 4] or [23, 24]): Expansion of the determinant as sum over the symmetric group, • Interpretation of this sum as the generating function of some set of combinatorial • objects which are equipped with some signed weight, Construction of an explicit weight– and sign–preserving bijection between the re- • spective combinatorial objects, maybe supported by the construction of a sign– reversing involution for certain objects. the electronic journal of combinatorics 8 (2001), #R16 1 Here, we will present another “method” of bijective proofs for determinant identitities, which involves the following steps: First, we replace the entries ai;j of the determinants by hλ i+j (where hm denotes i− • the m–th complete homogeneous function), Second, by the Jacobi–Trudi identity we transform the original determinant identity • into an equivalent identity for Schur functions, Third, we obtain a bijective proof for this equivalent identity by using the interpre- • tation of Schur functions in terms of nonintersecting lattice paths. -
Introduction to Linear Logic
Introduction to Linear Logic Beniamino Accattoli INRIA and LIX (École Polytechnique) Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 1 / 49 Outline 1 Informal introduction 2 Classical Sequent Calculus 3 Sequent Calculus Presentations 4 Linear Logic 5 Catching non-linearity 6 Expressivity 7 Cut-Elimination 8 Proof-Nets Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 2 / 49 Outline 1 Informal introduction 2 Classical Sequent Calculus 3 Sequent Calculus Presentations 4 Linear Logic 5 Catching non-linearity 6 Expressivity 7 Cut-Elimination 8 Proof-Nets Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 3 / 49 Quotation From A taste of Linear Logic of Philip Wadler: Some of the best things in life are free; and some are not. Truth is free. You may use a proof of a theorem as many times as you wish. Food, on the other hand, has a cost. Having baked a cake, you may eat it only once. If traditional logic is about truth, then Linear Logic is about food Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 4 / 49 Informally 1 Classical logic deals with stable truths: if A and A B then B but A still holds) Example: 1 A = ’Tomorrow is the 1st october’. 2 B = ’John will go to the beach’. 3 A B = ’If tomorrow is the 1st october then John will go to the beach’. So if tomorrow) is the 1st october, then John will go to the beach, But of course tomorrow will still be the 1st october. Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 5 / 49 Informally 2 But with money, or food, that implication is wrong: 1 A = ’John has (only) 5 euros’. -
Arxiv:1809.02384V2 [Math.OA] 19 Feb 2019 H Ruet Eaefloigaecmlctd N Oi Sntcerin Clear Not Is T It Saying So Worth and Is Complicated, It Are Following Hand
INVOLUTIVE OPERATOR ALGEBRAS DAVID P. BLECHER AND ZHENHUA WANG Abstract. Examples of operator algebras with involution include the op- erator ∗-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function alge- bras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators stud- ied by Garcia, Putinar, Wogen, Zhu, and others. We investigate the general theory of involutive operator algebras, and give many applications, such as a characterization of the symmetric operator algebras introduced in the early days of operator space theory. 1. Introduction An operator algebra for us is a closed subalgebra of B(H), for a complex Hilbert space H. Here we study operator algebras with involution. Examples include the operator ∗-algebras occurring in noncommutative differential geometry studied re- cently by Mesland, Kaad, Lesch, and others (see e.g. [26, 25, 10] and references therein), (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators studied by Garcia, Putinar, Wogen, Zhu, and many others (see [21] for a survey, or e.g. [22]). By an operator ∗-algebra we mean an operator algebra with an involution † † N making it a ∗-algebra with k[aji]k = k[aij]k for [aij ] ∈ Mn(A) and n ∈ . Here we are using the matrix norms of operator space theory (see e.g. [30]). This notion was first introduced by Mesland in the setting of noncommutative differential geometry [26], who was soon joined by Kaad and Lesch [25]. In several recent papers by these authors and coauthors they exploit operator ∗-algebras and involutive modules in geometric situations. -
Math 958–Topics in Discrete Mathematics Spring Semester 2018 TR 09:30–10:45 in Avery Hall (AVH) 351 1 Instructor
Math 958{Topics in Discrete Mathematics Spring Semester 2018 TR 09:30{10:45 in Avery Hall (AVH) 351 1 Instructor Dr. Tri Lai Assistant Professor Department of Mathematics University of Nebraska - Lincoln Lincoln, NE 68588-0130, USA Email: [email protected] Website: http://www.math.unl.edu/$\sim$tlai3/ Office: Avery Hall 339 Office Hours: By Appointment 2 Prerequisites Math 450. 3 Contacting me The best way to contact with me is by email, [email protected]. Please put \[MATH 958]" at the beginning of the title and make sure to include your whole name in your email. Using your official UNL email to contact me is strongly recommended. My office is Avery Hall 339. My office hours are by appoinment. 4 Course Description Bijective Combinatorics is a branch of combinatorics focusing on bijections between mathematical objects. The fact is that there many totally different objects in mathematics that are actually equinumerous, in this case, we always want to seek for a simple bijection between them. For example, the number of ways to divide that a convex (n + 2)-gon into triangles by non-intersecting diagonals is equal to the number of full binary trees with n + 1 leaves. Both objects are counted 1 2n by the well known Catalan number Cn = n+1 n . Do you know that there are more than two hundred (!!) mathematical objects are counted by the Catalan number? In this course, we will go over a number of such \Catalan objects" and investigate the beautiful bijections between them. One more example is the well-known theorem by Euler about integer partitions stating that the number of ways to write a positive integer n as the sum of distinct positive integers is equal to the number of ways to write n as the sum of odd positive integers. -
Finding Direct Partition Bijections by Two-Directional Rewriting Techniques
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 285 (2004) 151–166 www.elsevier.com/locate/disc Finding direct partition bijections by two-directional rewriting techniques Max Kanovich Department of Computer and Information Science, University of Pennsylvania, 3330 Walnut Street, Philadelphia, PA 19104, USA Received 7 May 2003; received in revised form 27 October 2003; accepted 21 January 2004 Abstract One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consists in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established in a systematic way from the ‘lattice properties’ of partition ideals, and how the desired bijections are computed by means of multiset rewriting, for a variety of combinatorial problems involving partitions. In particular, we fully characterizes all equinumerous partition ideals with ‘disjointly supported’ complements. This geometrical characterization is proved to automatically provide the desired bijection between partition ideals but in terms of the minimal elements of the order ÿlters, their complements. As a corollary, a new transparent proof, the ‘bijective’ one, is given for all equinumerous classes of the partition ideals of order 1 from the classical book “The Theory of Partitions” by G.Andrews. Establishing the required bijections involves two-directional reductions technique novel in the sense that forward and backward application of rewrite rules heads, respectively, for two di?erent normal forms (representing the two combinatorial types). It is well-known that non-overlapping multiset rules are con@uent. -
Orthogonal Involutions and Totally Singular Quadratic Forms In
Orthogonal involutions and totally singular quadratic forms in characteristic two A.-H. Nokhodkar Abstract We associate to every central simple algebra with involution of ortho- gonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution. Mathematics Subject Classification: 16W10, 16K20, 11E39, 11E04. 1 Introduction The theory of algebras with involution is closely related to the theory of bilinear forms. Assigning to every symmetric or anti-symmetric regular bilinear form on a finite-dimensional vector space V its adjoint involution induces a one- to-one correspondence between the similarity classes of bilinear forms on V and involutions of the first kind on the endomorphism algebra EndF (V ) (see [5, p. 1]). Under this correspondence several basic properties of involutions reflect analogous properties of bilinear forms. For example a symmetric or anti- symmetric bilinear form is isotropic (resp. anisotropic, hyperbolic) if and only if its adjoint involution is isotropic (resp. anisotropic, hyperbolic). Let (A, σ) be a totally decomposable algebra with orthogonal involution over a field F of characteristic two. In [3], a bilinear Pfister form Pf(A, σ) was associated to (A, σ), which determines the isotropy behaviour of (A, σ). It was shown that for every splitting field K of A, the involution σK on AK becomes adjoint to the bilinear form Pf(A, σ)K (see [3, (7.5)]). -
I-Rings with Involution
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOL-SAL OF ALGEBRA 38, 85-92 (1976) I-Rings with Involution Department of Mathematics, l.~&e~sity of Southern California, Los Angeles, California 90007 Communicated ly I. :V. Herstein Received September 18, 1974 The conditions of van Keumann regukity, x-reguiarity, and being an I-ring are placed on symmetric subrings of a ring with involution, and n-e determine when the whole ring must satisfy the same property. It is shown that any symmetric subring must satisfy any one of these properties if the whole ring does. In this paper, we continue our investigations begun in [6], in which we determined conditions on the symmetric elements of a ring witn involution that force the ring to be an I-ring. In the spirit of [S], the conditions of van Neumann reguhxrity, r-regularity, and being an I-ring are assumed for certain subrings generated by symmetric elements, to see if these conditions are then implied for the whole ring. Any of these conditions on the ring wiil restrict to the subrings under consideration, but we can only show that the I-ring condition extends up. In addition, we prove that r-regularity extends to the whole ring in the presence of a polynomial identity, and that the regularity condition cannot in general, be extended to the whole ring. Throughout this work, R will denote an associative ring with involution *. Let S = {.x E R i x:” = x> be the symmetric elements of R, 2” = {x L .x* i x E R] I the set of traces, and A’ = (xxx 1 x E RI, the set of norms. -
Lecture 8: Counting
Lecture 8: Counting Nitin Saxena ? IIT Kanpur 1 Generating functions (Fibonacci) P i 1 We had obtained φ(t) = i≥0 Fit = 1−t−t2 . We can factorize the polynomial, 1 − t − t2 = (1 − αt)(1 − βt), where α; β 2 C. p p 1+ 5 1− 5 Exercise 1. Show that α = 2 ; β = 2 . We can simplify the formula a bit more, 1 c c φ(t) = = 1 + 2 = c (1 + αt + α2t2 + ··· ) + c (1 + βt + β2t2 + ··· ) : 1 − t − t2 1 − αt 1 − βt 1 2 We got the second equality by putting c = p1 α and c = − p1 β. The final expression gives us an explicit 1 5 2 5 formula for the Fibonacci sequence, 1 n+1 n+1 Fn = p (α − β ) : 5 This is actually a very strong method and can be used for solving linear recurrences of the kind, Sn = a1Sn−1 + ··· + akSn−k. Here k; a1; a2; ··· ; ak are constants. Suppose φ(t) is the recurrence for Sn, then by the above method, k−1 b1 + b2t + ··· + bkt c1 ck φ(t) = 2 k = + ··· + : 1 − a1t − a2t − · · · − akt 1 − α1t 1 − αkt k k−1 Where α1; ··· ; αk are the roots (assumed distinct) of polynomial x − a1x − · · · − ak = 0 (replace t 1 by x ). This is known as the characteristic polynomial of the recurrence. Exercise 2. How are the coefficients b1; ··· ; bk or c1; ··· ; ck determined? nta conditions. Initial n n So we get Sn = c1α1 + ··· + ckαk . Note 1. For a linear recurrence if Fn and Gn are solutions then their linear combinations aFn + bGn are n n also solutions. -
Rational Catalan Combinatorics 1
Rational Catalan Combinatorics Drew Armstrong et al. University of Miami www.math.miami.edu/∼armstrong March 18, 2012 This talk will advertise a definition. Here is it. Definition Let x be a positive rational number written as x = a=(b − a) for 0 < a < b coprime. Then we define the Catalan number 1 a + b (a + b − 1)! Cat(x) := = : a + b a; b a!b! !!! Please note the a; b-symmetry. This talk will advertise a definition. Here is it. Definition Let x be a positive rational number written as x = a=(b − a) for 0 < a < b coprime. Then we define the Catalan number 1 a + b (a + b − 1)! Cat(x) := = : a + b a; b a!b! !!! Please note the a; b-symmetry. This talk will advertise a definition. Here is it. Definition Let x be a positive rational number written as x = a=(b − a) for 0 < a < b coprime. Then we define the Catalan number 1 a + b (a + b − 1)! Cat(x) := = : a + b a; b a!b! !!! Please note the a; b-symmetry. Special cases. When b = 1 mod a ::: I Eug`eneCharles Catalan (1814-1894) (a < b) = (n < n + 1) gives the good old Catalan number n 1 2n + 1 Cat(n) = Cat = : 1 2n + 1 n I Nicolaus Fuss (1755-1826) (a < b) = (n < kn + 1) gives the Fuss-Catalan number n 1 (k + 1)n + 1 Cat = : (kn + 1) − n (k + 1)n + 1 n Special cases. When b = 1 mod a ::: I Eug`eneCharles Catalan (1814-1894) (a < b) = (n < n + 1) gives the good old Catalan number n 1 2n + 1 Cat(n) = Cat = : 1 2n + 1 n I Nicolaus Fuss (1755-1826) (a < b) = (n < kn + 1) gives the Fuss-Catalan number n 1 (k + 1)n + 1 Cat = : (kn + 1) − n (k + 1)n + 1 n Special cases. -
The Involution Tool for Accurate Digital Timingand Power Analysis Daniel Ohlinger, Jürgen Maier, Matthias Függer, Ulrich Schmid
The Involution Tool for Accurate Digital Timingand Power Analysis Daniel Ohlinger, Jürgen Maier, Matthias Függer, Ulrich Schmid To cite this version: Daniel Ohlinger, Jürgen Maier, Matthias Függer, Ulrich Schmid. The Involution Tool for Accu- rate Digital Timingand Power Analysis. PATMOS 2019 - 29th International Symposium on Power and Timing Modeling, Optimization and Simulation, Jul 2019, Rhodes, Greece. 10.1109/PAT- MOS.2019.8862165. hal-02395242 HAL Id: hal-02395242 https://hal.inria.fr/hal-02395242 Submitted on 5 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Involution Tool for Accurate Digital Timing and Power Analysis Daniel Ohlinger¨ ∗,Jurgen¨ Maier∗, Matthias Fugger¨ y, Ulrich Schmid∗ ∗ECS Group, TU Wien [email protected], jmaier, s @ecs.tuwien.ac.at f g yCNRS & LSV, ENS Paris-Saclay, Universite´ Paris-Saclay & Inria [email protected] Abstract—We introduce the prototype of a digital timing sim- in(t) ulation and power analysis tool for integrated circuit (Involution Tool) which employs the involution delay model introduced by t Fugger¨ et al. at DATE’15. Unlike the pure and inertial delay out(t) models typically used in digital timing analysis tools, the involu- tion model faithfully captures pulse propagation. -
On a Class of Inverse Problems for a Heat Equation with Involution Perturbation
On a class of inverse problems for a heat equation with involution perturbation Nasser Al-Salti∗y, Mokhtar Kiranez and Berikbol T. Torebekx Abstract A class of inverse problems for a heat equation with involution pertur- bation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed. Keywords: Inverse Problems, Heat Equation, Involution Perturbation. 2000 AMS Classification: 35R30; 35K05; 39B52. 1. Introduction Differential equations with modified arguments are equations in which the un- known function and its derivatives are evaluated with modifications of time or space variables; such equations are called in general functional differential equa- tions. Among such equations, one can single out equations with involutions [?]. 1.1. Definition. [?, ?] A function α(x) 6≡ x that maps a set of real numbers, Γ, onto itself and satisfies on Γ the condition α (α(x)) = x; or α−1(x) = α(x) is called an involution on Γ: Equations containing involution are equations with an alternating deviation (at x∗ < x being equations with advanced, and at x∗ > x being equations with delay, where x∗ is a fixed point of the mapping α(x)). Equations with involutions have been studied by many researchers, for exam- ple, Ashyralyev [?, ?], Babbage [?], Przewoerska-Rolewicz [?, ?, ?, ?, ?, ?, ?], Aftabizadeh and Col. [?], Andreev [?, ?], Burlutskayaa and Col. [?], Gupta [?, ?, ?], Kirane [?], Watkins [?], and Wiener [?, ?, ?, ?]. Spectral problems and inverse problems for equations with involutions have received a lot of attention as well, see for example, [?, ?, ?, ?, ?], and equations with a delay in the space variable have been the subject of many research papers, see for example, [?, ?].