10.2 Robinson-Schensted-Knuth (RSK) Correspondence
6 min read•july 30, 2024
The Robinson-Schensted-Knuth (RSK) correspondence is a powerful tool in algebraic combinatorics. It links permutations to pairs of standard , revealing hidden patterns and structures in seemingly random sequences of numbers.
RSK ties into the broader study of tableaux combinatorics and Schur functions. By connecting permutations to tableaux, it bridges different areas of math, offering insights into symmetric functions, representation theory, and permutation statistics.
RSK correspondence
Bijection between permutations and pairs of standard Young tableaux
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- The RSK correspondence establishes a one-to-one correspondence () between permutations of the set {1, 2, ..., n} and pairs of standard Young tableaux of the same shape with n boxes
- Each permutation is uniquely associated with a pair of standard Young tableaux, called the insertion tableau (P) and the recording tableau (Q)
- The insertion tableau (P) results from applying the RSK insertion algorithm to the permutation, while the recording tableau (Q) tracks the order of element insertion
- The shape of the resulting Young tableaux, determined by the length of the longest increasing subsequence in the permutation, is a of n (3, 2, 1)
- RSK correspondence preserves properties of permutations, such as the length of the longest increasing subsequence (4, 5, 6) and the length of the longest decreasing subsequence (6, 5, 4)
Properties preserved by RSK correspondence
- The length of the longest increasing subsequence in a permutation equals the number of rows in the corresponding Young tableaux obtained from RSK
- The length of the longest decreasing subsequence in a permutation equals the number of columns in the corresponding Young tableaux
- RSK correspondence provides a combinatorial proof of the Cauchy identity, relating Schur functions and complete homogeneous symmetric functions
- RSK correspondence helps study the distribution of permutation statistics, such as the length of the longest increasing subsequence and the number of inversions (pairs of elements in the wrong order)
- RSK correspondence connects to representation theory by relating irreducible representations of the to Young tableaux and Schur functions
- RSK correspondence generalizes to other combinatorial objects (words, matrices), leading to the study of RSK-type algorithms and their properties
Constructing tableaux
Insertion tableau (P) construction
- To construct the insertion tableau (P), start with an empty Young diagram and insert the permutation elements one by one using the RSK insertion algorithm
- RSK insertion algorithm:
- If the current element is greater than all elements in the first row, append it to the end of the first row
- Otherwise, find the leftmost element in the first row greater than the current element, replace it with the current element, and bump the replaced element to the next row
- Recursively repeat the process for the bumped element in the next row until the element finds its place in the tableau
- Example: For the permutation (3, 1, 4, 2), the insertion tableau (P) would be constructed as follows:
- Insert 3: [3]
- Insert 1: [1, 3]
- Insert 4: [1, 3, 4]
- Insert 2: [1, 2, 4] [3]
Recording tableau (Q) construction
- To construct the recording tableau (Q), start with an empty Young diagram and add a box labeled with the position of each element in the permutation as it is inserted into the insertion tableau (P)
- The recording tableau (Q) keeps track of the order in which the elements are inserted into the insertion tableau (P)
- The resulting insertion tableau (P) and recording tableau (Q) form a pair of standard Young tableaux of the same shape
- Example: For the permutation (3, 1, 4, 2), the recording tableau (Q) would be: [1, 2, 3] [4]
Recovering permutations
Reverse RSK algorithm
- Given a pair of standard Young tableaux (P, Q) obtained from the RSK correspondence, the original permutation can be recovered using the reverse RSK algorithm
- Reverse RSK algorithm:
- Start with the rightmost box in the bottom row of the recording tableau (Q) and find the corresponding box with the same label in the insertion tableau (P)
- Remove the element from the insertion tableau (P) and append it to the beginning of the permutation
- Perform the reverse bumping procedure: if the removed element was at the end of its row, stop; otherwise, find the largest element in the previous row smaller than the removed element, and replace it with the removed element
- Recursively repeat the process for the bumped element in the previous row until no more bumping is possible
- Continue the process for each box in the recording tableau (Q), moving from right to left and bottom to top, until all boxes have been processed
- The resulting sequence of elements obtained from the insertion tableau (P) is the original permutation
Example of recovering permutation
- Given the following pair of standard Young tableaux (P, Q): P: [1, 2, 4] [3] Q: [1, 2, 3] [4]
- Recover the original permutation:
- Start with the rightmost box in the bottom row of Q (labeled 4) and find the corresponding box in P (containing 3)
- Remove 3 from P and append it to the beginning of the permutation: (3)
- Perform reverse bumping: 3 is at the end of its row, so stop
- Move to the next box in Q (labeled 3) and find the corresponding box in P (containing 4)
- Remove 4 from P and append it to the beginning of the permutation: (4, 3)
- Perform reverse bumping: replace 4 with 2 in the first row of P
- Move to the next box in Q (labeled 2) and find the corresponding box in P (containing 2)
- Remove 2 from P and append it to the beginning of the permutation: (2, 4, 3)
- Perform reverse bumping: 2 is at the end of its row, so stop
- Move to the final box in Q (labeled 1) and find the corresponding box in P (containing 1)
- Remove 1 from P and append it to the beginning of the permutation: (1, 2, 4, 3)
- The resulting permutation is (1, 2, 4, 3), which was the original permutation used to construct the tableaux
Properties and applications of RSK
Connections to other areas of mathematics
- RSK correspondence is a fundamental tool in algebraic combinatorics with numerous applications in various areas of mathematics
- RSK correspondence relates irreducible representations of the symmetric group to Young tableaux and Schur functions, establishing a connection to representation theory
- RSK correspondence provides a combinatorial proof of the Cauchy identity, which relates Schur functions and complete homogeneous symmetric functions, linking it to symmetric function theory
- RSK correspondence can be generalized to other combinatorial objects, such as words and matrices, leading to the study of RSK-type algorithms and their properties in combinatorics and algebra
Applications in combinatorics and permutation statistics
- RSK correspondence helps analyze the properties of permutations and their statistics
- The length of the longest increasing subsequence in a permutation equals the number of rows in the corresponding Young tableaux obtained from RSK
- The length of the longest decreasing subsequence in a permutation equals the number of columns in the corresponding Young tableaux
- RSK correspondence can be used to study the distribution of certain permutation statistics, such as the length of the longest increasing subsequence and the number of inversions (pairs of elements in the wrong order)
- Understanding the properties of permutations through RSK correspondence helps in solving various combinatorial problems and understanding the structure of permutations (pattern avoidance, permutation patterns)
Key Terms to Review (16)
Bijection: A bijection is a function that establishes a one-to-one correspondence between two sets, meaning that every element in the first set is paired with exactly one unique element in the second set, and vice versa. This property is crucial in various areas as it allows for counting, mapping, and comparing the sizes of different sets effectively, providing a foundation for concepts like enumeration, combinatorial structures, and relationships between different mathematical objects.
Daniel Schensted: Daniel Schensted is a prominent mathematician known for his significant contributions to combinatorics and representation theory, particularly through the development of the Robinson-Schensted-Knuth (RSK) correspondence. This correspondence creates a powerful link between permutations and combinatorial objects such as Young tableaux, allowing for deeper insights into algebraic structures and their applications in various mathematical fields.
Donald Knuth: Donald Knuth is a renowned computer scientist, mathematician, and author best known for his contributions to algorithms and typesetting. He developed the influential typesetting system TeX and introduced the concept of analysis of algorithms, which lays the groundwork for understanding computational complexity. His work is crucial in combinatorial algorithms, particularly in relation to the Robinson-Schensted-Knuth correspondence and its applications in algorithmic design.
Dual Equivalence: Dual equivalence refers to a relationship between two combinatorial objects that allows for a correspondence in their structures, particularly in the context of the Robinson-Schensted-Knuth (RSK) correspondence. This concept highlights how certain operations or transformations applied to one object can be mirrored in another, revealing deeper connections and symmetries within combinatorial theory.
G. de b. robinson: The term g. de b. robinson refers to a specific method of analyzing the structure and behavior of tableaux in the context of the Robinson-Schensted-Knuth (RSK) correspondence, which is a fundamental concept in combinatorial representation theory. This method helps in understanding how permutations can be represented as pairs of standard Young tableaux and provides insights into various algebraic and combinatorial properties, such as longest increasing subsequences and the shape of the tableaux.
Insertion procedure: The insertion procedure is a fundamental algorithm used in the context of the Robinson-Schensted-Knuth (RSK) correspondence that involves inserting elements into a growing sequence, creating a standard Young tableau. This process highlights how permutations can be represented combinatorially and connects to important concepts like shape and depth in the theory of tableaux, reflecting the structural relationships among them.
Knuth Transformation: The Knuth Transformation is a combinatorial operation that transforms a sequence into another sequence while preserving certain combinatorial properties. This transformation plays a crucial role in the context of the Robinson-Schensted-Knuth (RSK) Correspondence by facilitating the construction of a new sequence from an original one, allowing for the exploration of relationships between permutations and Young tableaux.
Partition: In combinatorics, a partition is a way of breaking a set of objects into non-empty subsets where the order of subsets does not matter. Partitions are crucial for understanding how to count different configurations, and they connect to concepts such as counting methods, combinatorial identities, and representation theory.
Probability theory: Probability theory is the branch of mathematics that deals with the analysis of random phenomena and the likelihood of various outcomes. It provides a framework for quantifying uncertainty and is essential in many areas, including statistics, finance, and combinatorial structures. This theory forms the backbone of numerous concepts, such as random variables, distributions, and expected values, which are integral to understanding complex combinatorial relationships.
Robinson-Schensted Algorithm: The Robinson-Schensted algorithm is a combinatorial algorithm that establishes a correspondence between permutations and pairs of standard Young tableaux of the same shape. This algorithm is significant as it not only provides a way to encode permutations but also highlights the deep connections between algebraic and combinatorial structures, playing a key role in representation theory and the study of symmetric functions.
Semi-standard young tableau: A semi-standard young tableau is a way to fill a Young diagram with positive integers that weakly increase across each row and strictly increase down each column. This structure is crucial in combinatorics, particularly in representing the RSK correspondence, where each tableau corresponds to a specific permutation and provides insights into the representation theory of symmetric groups.
Skew tableau: A skew tableau is a type of Young tableau that is formed by placing numbers in a partially filled shape, where the shape consists of squares removed from a rectangular array. This structure allows for specific arrangements of numbers, following the rules of increasing sequences both across rows and down columns. Skew tableaux are particularly important in understanding combinatorial representations and play a crucial role in the RSK correspondence, linking them to the representation theory of symmetric groups.
Standard Young Tableau: A Standard Young Tableau is a way of filling the boxes of a Young diagram with integers such that the numbers in each row and each column are strictly increasing. This concept is key in combinatorics, as it relates to counting problems and representation theory, connecting to the study of symmetric functions, combinatorial algorithms, and the interplay between algebra and geometry.
Symmetric group: The symmetric group is a fundamental concept in abstract algebra that consists of all possible permutations of a finite set of elements, forming a group under the operation of composition. This group captures the notion of rearranging objects and plays a crucial role in combinatorics, representation theory, and many other areas of mathematics.
Tableau shape: A tableau shape is a specific arrangement of boxes in a Young tableau, typically defined by the number of boxes in each row, which must be non-increasing from top to bottom. This geometric structure serves as a fundamental aspect in the study of combinatorial representations, particularly in relation to the RSK correspondence, where the shape of the tableau plays a critical role in understanding how permutations can be represented and transformed.
Young tableaux: Young tableaux are combinatorial objects that represent ways to fill the boxes of a Young diagram with numbers, subject to certain rules. They provide a way to visualize and study the relationships between different representations of symmetric groups, and they have applications in various areas, including algebra and geometry, as well as connections to symmetric functions and representation theory.