youngTableau -- constructor for type YoungTableau

Usage:

youngTableau(lam, mu, entryList)

youngTableau((lam,mu),entryList)

youngTableau(lam, entryList)
Inputs:
- lam, an instance of the type Partition, the outer shape $\lambda$.
- mu, an instance of the type Partition, the inner shape $\mu$. If not given, then it is assumed to be the $0$ partition.
- entryList, a list, the filling of the boxes. If not given, then box entries are assumed to be the empty string "".
Outputs:
- T, an instance of the type YoungTableau, a skew Young tableau of shape $\lambda/\mu$ with the given filling.
Consequences:
- The list of entries has length equal to $\sum_{i=1}^{\ell(\lambda)}|\lambda_i-\mu_i|$. E.g., if $\lambda=(2)$ and $\mu=(5)$, then the entry list must have length $3$.
- None of the entries are null.

Description

i1 : lam = new Partition from {4,3,2}

o1 = Partition{4, 3, 2}

o1 : Partition

i2 : mu = new Partition from {3,1}

o2 = Partition{3, 1}

o2 : Partition

i3 : entryList = {1,2,3,3,9}

o3 = {1, 2, 3, 3, 9}

o3 : List

i4 : T = youngTableau(lam,mu,entryList)

                 ┌───┐
o4 =             │ 1 │
         ┌───┬───┼───┘
         │ 2 │ 3 │    
     ┌───┼───┼───┘    
     │ 3 │ 9 │        
     └───┴───┘        

o4 : YoungTableau

We may construct tableaux with any compositions.

i5 : youngTableau(new Partition from {3,5,1}, new Partition from {0,1}, {7,"&",4,2,"g","u",6,0})

     ┌───┬───┬───┐        
o5 = │ 7 │ & │ 4 │        
     └───┼───┼───┼───┬───┐
         │ 2 │ g │ u │ 6 │
     ┌───┼───┴───┴───┴───┘
     │ 0 │                
     └───┘                

o5 : YoungTableau

The shapes may have negative parts. In this case, a vertical line is drawn by net to indicate that negative parts are to the left.

i6 : youngTableau(new Partition from {3,-3,-1}, new Partition from {-2,-4,-1})

             ┌───┬───║───┬───┬───┐
o6 =         │   │   ║   │   │   │
     ┌───┐   └───┴───║───┴───┴───┘
     │   │           ║            
     └───┘           ║            
                     ║            
                     ║            

o6 : YoungTableau

If any $\lambda_i<\mu_i$, then the boxes in that row are drawn shaded.

i7 : youngTableau(new Partition from {-2,-4,2}, new Partition from {1,-1,-1}, {1,2,3,4,5,6,7,8,9})

             ┌───┬───║───┐    
o7 =         │░1░│░2░║░3░│    
     ┌───┬───┼───┼───║───┘    
     │░4░│░5░│░6░│   ║        
     └───┴───┴───┼───║───┬───┐
                 │ 7 ║ 8 │ 9 │
                 └───║───┴───┘

o7 : YoungTableau

The inner shape may be drawn by calling drawInnerShape.

i8 : T'' = youngTableau(new Partition from {5,4,-1}, new Partition from {2,4,-3})

                 ║       ┌───┬───┬───┐
o8 =             ║       │   │   │   │
                 ║       └───┴───┴───┘
                 ║                    
     ┌───┬───┐   ║                    
     │   │   │   ║                    
     └───┴───┘   ║                    

o8 : YoungTableau

i9 : drawInnerShape true 

o9 = true

i10 : T''

                  ║       ┌───┬───┬───┐
o10 =             ║ ■   ■ │   │   │   │
                  ║       └───┴───┴───┘
                  ║ ■   ■   ■   ■      
      ┌───┬───┐   ║                    
      │   │   │ ■ ║                    
      └───┴───┘   ║                    

o10 : YoungTableau

i11 : drawInnerShape false 

o11 = false

i12 : T''

                  ║       ┌───┬───┬───┐
o12 =             ║       │   │   │   │
                  ║       └───┴───┴───┘
                  ║                    
      ┌───┬───┐   ║                    
      │   │   │   ║                    
      └───┴───┘   ║                    

o12 : YoungTableau

The filling may be of any class (besides null objects), including other tableaux!

i13 : youngTableau(lam,mu,for i from 1 to size T list randomTabloid lam)

                                                                  ┌───────────────────┐
o13 =                                                             │ ┌───────────────┐ │
                                                                  │ │ 6   4   7   9 │ │
                                                                  │ ├───────────┬───┘ │
                                                                  │ │ 2   3   1 │     │
                                                                  │ ├───────┬───┘     │
                                                                  │ │ 5   8 │         │
                                                                  │ └───────┘         │
                          ┌───────────────────┬───────────────────┼───────────────────┘
                          │ ┌───────────────┐ │ ┌───────────────┐ │                    
                          │ │ 5   4   1   7 │ │ │ 8   3   4   6 │ │                    
                          │ ├───────────┬───┘ │ ├───────────┬───┘ │                    
                          │ │ 3   8   2 │     │ │ 1   5   9 │     │                    
                          │ ├───────┬───┘     │ ├───────┬───┘     │                    
                          │ │ 9   6 │         │ │ 2   7 │         │                    
                          │ └───────┘         │ └───────┘         │                    
      ┌───────────────────┼───────────────────┼───────────────────┘                    
      │ ┌───────────────┐ │ ┌───────────────┐ │                                        
      │ │ 6   4   7   9 │ │ │ 1   2   3   5 │ │                                        
      │ ├───────────┬───┘ │ ├───────────┬───┘ │                                        
      │ │ 3   1   8 │     │ │ 9   6   7 │     │                                        
      │ ├───────┬───┘     │ ├───────┬───┘     │                                        
      │ │ 5   2 │         │ │ 8   4 │         │                                        
      │ └───────┘         │ └───────┘         │                                        
      └───────────────────┴───────────────────┘                                        

o13 : YoungTableau

Ways to use youngTableau:

youngTableau(Partition)
youngTableau(Partition,List)
youngTableau(Partition,Partition)
youngTableau(Partition,Partition,List)
youngTableau(Sequence,List)
youngTableau(Tabloid)

For the programmer

The object youngTableau is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Tableaux/documentation.m2:1594:0.

youngTableau -- constructor for type YoungTableau

Description

See also

Ways to use youngTableau:

For the programmer