# Affine Type - Explore the Science & Experts | ideXlab

## Affine Type

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### Pavlo Pylyavskyy – One of the best experts on this subject based on the ideXlab platform.

• ##### two row w graphs in Affine Type a
Co-Authors: Dongkwan Kim, Pavlo Pylyavskyy
Abstract:

Abstract For Affine symmetric groups we construct finite W-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of purely combinatorial constructions of finite W-graphs in an Affine Type. We compare our construction with quotients of periodic W-graphs defined by Lusztig. Under certain positivity assumption on the latter the two are shown to be isomorphic.

• ##### two row w graphs in Affine Type a
arXiv: Combinatorics, 2019
Co-Authors: Dongkwan Kim, Pavlo Pylyavskyy
Abstract:

For Affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an Affine Type. We compare our construction with quotients of periodic $W$-graphs defined by Lusztig. Under certain positivity assumption on the latter the two are shown to be isomorphic.

• ##### Matrix-Ball Construction of Affine Robinson–Schensted correspondence
Selecta Mathematica, 2018
Co-Authors: Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina
Abstract:

In his study of Kazhdan–Lusztig cells in Affine Type A , Shi has introduced an Affine analog of Robinson–Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi’s algorithm. As a byproduct, we also give a way to realize the Affine correspondence via the usual Robinson–Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between the extended Affine symmetric group and collection of triples $$(P, Q, \rho )$$ ( P , Q , ρ ) where P and Q are tabloids and $$\rho$$ ρ is a dominant weight. The weights $$\rho$$ ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.

### Vladimir Zamdzhiev – One of the best experts on this subject based on the ideXlab platform.

• ##### Semantics for First-Order Affine Inductive Data Types via Slice Categories
, 2020
Abstract:

Affine Type systems are substructural Type systems where copying of information is restricted, but discarding of information is permissible at all Types. Such Type systems are well-suited for describing quantum programming languages, because copying of quantum information violates the laws of quantum mechanics. In this paper, we consider a first-order Affine Type system with inductive data Types and present a novel categorical semantics for it. The most challenging aspect of this interpretation comes from the requirement to construct appropriate discarding maps for our data Types which might be defined by mutual/nested recursion. We show how to achieve this for all Types by taking models of a first-order linear Type system whose atomic Types are discardable and then presenting an additional Affine interpretation of Types within the slice category of the model with the tensor unit. We present some concrete categorical models for the language ranging from classical to quantum. Finally, we discuss potential ways of dualising and extending our methods and using them for interpreting coalgebraic and lazy data Types.

• ##### Computational Adequacy for Substructural Lambda Calculi
, 2020
Abstract:

Substructural Type systems, such as Affine (and linear) Type systems, are Type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum programming. We describe one linear and one Affine Type systems and we formulate abstract categorical models for both of them which are sound and computationally adequate. We also show, under basic assumptions, that interpreting lambda abstractions via a monoidal closed structure (a popular method for linear Type systems) necessarily leads to degenerate and inadequate models for call-by-value Affine Type systems, so we avoid doing this in our categorical treatment, where a solution to this problem is clearly identified. Our categorical models are more general than linear/non-linear models used to study linear logic and we present a homogeneous categorical account of both linear and Affine Type systems in a call-by-value setting. We also give examples with many concrete models, including classical and quantum ones.

• ##### Quantum Programming with Inductive DataTypes: Causality and Affine Type Theory
, 2020
Co-Authors: Romain Péchoux, Simon Perdrix, Mathys Rennela, Vladimir Zamdzhiev
Abstract:

Inductive dataTypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive dataTypes may be added to the quantum progprogramming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semantic treatment of user-defined inductive dataTypes in quantum programming. We also show our denotational interpretation is invariant with respect to big-step reduction, thereby establishing another novel result for quantum programming. Compared to classical programming, this property is considerably more difficult to prove and we demonstrate its usefulness by showing how it immediately implies computational adequacy at all Types. To further cement our results, our semantics is entirely based on a physically natural model of von Neumann algebras, which are mathematical structures used by physicists to study quantum mechanics.

### Romain Péchoux – One of the best experts on this subject based on the ideXlab platform.

• ##### FLOPS – Polynomial time over the reals with parsimony
Functional and Logic Programming, 2020
Co-Authors: Emmanuel Hainry, Damiano Mazza, Romain Péchoux
Abstract:

We provide a characterization of Ko’s class of polynomial time computable functions over real numbers. This characterization holds for a stream based language using a parsimonious Type discipline, a variant of propositional linear logic. We obtain a first characterization of polynomial time computations over the reals on a higher-order functional language using a linear/Affine Type system.

• ##### Quantum Programming with Inductive DataTypes: Causality and Affine Type Theory
, 2020
Co-Authors: Romain Péchoux, Simon Perdrix, Mathys Rennela, Vladimir Zamdzhiev
Abstract:

Inductive dataTypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive dataTypes may be added to the quantum programming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semantic treatment of user-defined inductive dataTypes in quantum programming. We also show our denotational interpretation is invariant with respect to big-step reduction, thereby establishing another novel result for quantum programming. Compared to classical programming, this property is considerably more difficult to prove and we demonstrate its usefulness by showing how it immediately implies computational adequacy at all Types. To further cement our results, our semantics is entirely based on a physically natural model of von Neumann algebras, which are mathematical structures used by physicists to study quantum mechanics.

• ##### Polynomial time over the reals with parsimony
, 2020
Co-Authors: Emmanuel Hainry, Damiano Mazza, Romain Péchoux
Abstract:

We provide a characterization of Ko’s class of polynomial time computable functions over real numbers. This characterization holds for a stream based language using a parsimonious Type discipline, a variant of propositional linear logic. We obtain a first characterization of polynomial time computations over the reals on a higher-order functional language using a linear/Affine Type system.

### Peter J. Mcnamara – One of the best experts on this subject based on the ideXlab platform.

• ##### Representations of Khovanov–Lauda–Rouquier algebras III: symmetric Affine Type
Mathematische Zeitschrift, 2017
Co-Authors: Peter J. Mcnamara
Abstract:

We develop the homological theory of KLR algebras of symmetric Affine Type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.

• ##### representations of khovanov lauda rouquier algebras iii symmetric Affine Type
arXiv: Representation Theory, 2014
Co-Authors: Peter J. Mcnamara
Abstract:

We develop the homological theory of KLR algebras of symmetric Affine Type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.

### Dongkwan Kim – One of the best experts on this subject based on the ideXlab platform.

• ##### two row w graphs in Affine Type a
For Affine symmetric groups we construct finite $W$-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of examples of finite $W$-graphs in an Affine Type. We compare our construction with quotients of periodic $W$-graphs defined by Lusztig. Under certain positivity assumption on the latter the two are shown to be isomorphic.