pieri -- computes a matrix representation for a Pieri inclusion of representations of a general linear group

Usage:

pieri(mu, boxes, P)

pieri(mu, boxes, P, Convention => "Row" | "Filling" | "Weyl")
Inputs:
- mu, a list, a partition $(\mu_1, \ldots, \mu_r)$ where $\mu_i$ is the number of boxes in the $i$th row
- boxes, a list, a list of rows from which to remove boxes (boxes are always removed from the end of the row). This specifies which map of $GL(V)$ representations we want. The row indices start from 1 and not 0, and this must specify a horizontal strip in $\mu$ (see description below).
- P, a polynomial ring, a polynomial ring over a field $K$ in $n$ variables
Optional inputs:
- Convention => a string, default value "Row", , the tableau basis convention used to label the source $S_\mu V$ and target $S_\lambda V$. One of "Row" (default; PieriMaps row-form SSYT basis), "Filling" (SchurFunctors column-form Filling basis on both sides), or "Weyl" (data-equivalent to Row, but the bases are interpreted as WeylFillings on the divided-power side). The strip factor $(\mathrm{Sym}^k V)$ stays in P regardless.
Outputs:
- a matrix, If $K$ has characteristic 0, then given the partition $\mu$ and the partition $\mu'$ obtained from $\mu$ by removing a single box, there is a unique (up to nonzero scalar) $GL(V)$-equivariant inclusion $S_\mu V \to V \otimes S_{\mu'} V$, where $S_\mu$ refers to the irreducible representation of $GL(V)$ with highest weight $\mu$. This can be extended uniquely to a map of $P = \mathrm{Sym}(V)$-modules $P \otimes S_\mu V \to P \otimes S_{\mu'} V$. This method computes the matrix representation for the composition of maps that one obtains by iterating this procedure of removing boxes, i.e., the final output is a $GL(V)$-equivariant map $P \otimes S_\mu V \to P \otimes S_\lambda V$ where $\lambda$ is the partition obtained from $\mu$ by deleting a box from row $\mathrm{boxes}_0$, a box from row $\mathrm{boxes}_1$, etc. If $K$ has positive characteristic, then the corresponding map is calculated over $\mathbb{Q}$, lifted to a $\mathbb{Z}$-form of the representation which has the property that the map has a torsion-free cokernel over $\mathbb{Z}$, and then the coefficients are reduced to $K$.

Description

Convention: the partition $(d)$ represents the $d$th symmetric power, while the partition $(1, \ldots, 1)$ represents the $d$th exterior power. Using the notation from the output, $\mu/\lambda$ must be a horizontal strip. Precisely, this means that $\lambda_i \geq \mu_{i+1}$ for all $i$. If this condition is not satisfied, the program throws an error because a nonzero equivariant map of the desired form will not exist.

The Convention option re-expresses the same equivariant map in different tableau bases on the source $S_\mu V$ and target $S_\lambda V$. In "Row" (default), the bases are PM-style SSYT (rows weakly increasing). In "Filling", they are SchurFunctors-style column-form Fillings (the matrix is obtained by post-hoc basis change with the equivariant iso pmToFilling). In "Weyl", the matrix data is the same as "Row" but is interpreted on the WeylFilling (divided-power) side. All three give matrices of the same rank.

i1 : pieri({3,1}, {1}, QQ[a,b,c]) -- removes the last box from row 1 of the partition {3,1}

o1 = | 3a 0  b  0  c  0  0  0  0     0 0   0  0  0  0  |
     | 0  3a 0  b  0  c  0  0  0     0 0   0  0  0  0  |
     | 0  0  2a 0  0  0  2b 0  c     0 0   0  0  0  0  |
     | 0  0  0  2a 0  0  0  2b 0     c 0   0  0  0  0  |
     | 0  0  0  0  2a 0  0  0  b     0 2c  0  0  0  0  |
     | 0  0  0  0  0  2a 0  0  0     b 0   2c 0  0  0  |
     | 0  0  0  0  0  0  0  a  -1/2a 0 0   0  3b c  0  |
     | 0  0  0  0  0  0  0  0  0     a -2a 0  0  2b 2c |

                      8               15
o1 : Matrix (QQ[a..c])  <-- (QQ[a..c])

i2 : res coker oo -- resolve this map

               8               15               10               3
o2 = (QQ[a..c])  <-- (QQ[a..c])   <-- (QQ[a..c])   <-- (QQ[a..c])
                                                        
     0               1                2                3

o2 : Complex

i3 : betti oo -- check that the resolution is pure

            0  1  2 3
o3 = total: 8 15 10 3
         0: 8 15  . .
         1: .  . 10 .
         2: .  .  . 3

o3 : BettiTally

i4 : P = QQ[a,b,c];

i5 : Mrow = pieri({2,1}, {1}, P, Convention => "Row")

o5 = | 2a 0  b 0 c  0 0  0 |
     | 0  2a 0 b 0  c 0  0 |
     | 0  0  0 a -a 0 2b c |

             3      8
o5 : Matrix P  <-- P

i6 : P = QQ[a,b,c];

i7 : Mfil = pieri({2,1}, {1}, P, Convention => "Filling")

o7 = | 3/2a 3/2b c     0    1/2c 0    0    0    |
     | 0    0    1/2b  3/2a b    3/2c 0    0    |
     | 0    0    -1/2a 0    1/2a 0    3/2b 3/2c |

             3      8
o7 : Matrix P  <-- P

i8 : P = QQ[a,b,c];

i9 : Mwey = pieri({2,1}, {1}, P, Convention => "Weyl")

o9 = | 2a 0  b 0 c  0 0  0 |
     | 0  2a 0 b 0  c 0  0 |
     | 0  0  0 a -a 0 2b c |

             3      8
o9 : Matrix P  <-- P

The Row and Weyl matrices share raw data (only the source/target bases are reinterpreted on the divided-power side); the Filling matrix is obtained by post-hoc basis change. All three have the same rank.

i10 : P = QQ[a,b,c];

i11 : Mrow = pieri({2,1}, {1}, P, Convention => "Row");

              3      8
o11 : Matrix P  <-- P

i12 : Mfil = pieri({2,1}, {1}, P, Convention => "Filling");

              3      8
o12 : Matrix P  <-- P

i13 : Mwey = pieri({2,1}, {1}, P, Convention => "Weyl");

              3      8
o13 : Matrix P  <-- P

i14 : rank Mrow, rank Mfil, rank Mwey

o14 = (3, 3, 3)

o14 : Sequence

The Betti table of the cokernel is the same in every convention (the resolutions are isomorphic up to an equivariant change of basis):

i15 : P = QQ[a,b,c];

i16 : betti res coker pieri({2,1}, {1}, P, Convention => "Row")

             0 1 2 3
o16 = total: 3 8 6 1
          0: 3 8 6 .
          1: . . . 1

o16 : BettiTally

i17 : P = QQ[a,b,c];

i18 : betti res coker pieri({2,1}, {1}, P, Convention => "Filling")

             0 1 2 3
o18 = total: 3 8 6 1
          0: 3 8 6 .
          1: . . . 1

o18 : BettiTally

Multi-box horizontal strips: boxes can list more than one row index (with repeats), and the result is the composition of single-box Pieri inclusions. The same row index can appear repeatedly when row 1 has enough boxes; entries become degree-$d$ polynomials.

i19 : P = QQ[a,b,c];

i20 : pieri({3,1}, {1,1}, P)         -- removes 2 boxes from row 1; entries are deg 2

o20 = | 6a2 0   4ab 0   4ac  0   2b2 0   2bc  0   2c2  0   0   0   0   |
      | 0   6a2 0   4ab 0    4ac 0   2b2 0    2bc 0    2c2 0   0   0   |
      | 0   0   0   2a2 -2a2 0   0   4ab -2ab 2ac -4ac 0   6b2 4bc 2c2 |

              3      15
o20 : Matrix P  <-- P

i21 : P = QQ[a,b,c];

i22 : pieri({3,2,1}, {1,3}, P)       -- removes one box from row 1, then row 3

o22 = | 3ac  0      bc  0      c2  0      0      0      |
      | -3ab 3/2ac  -b2 1/2bc  -bc 1/2c2  0      0      |
      | 0    -3/2ab 0   -1/2b2 0   -1/2bc 0      0      |
      | 6a2  0      2ab ac     0   0      bc     1/2c2  |
      | 0    3/2a2  0   0      ab  1/2ac  -1/2b2 -1/4bc |
      | 0    0      0   1/2a2  -a2 0      1/2ab  1/4ac  |

              6      8
o22 : Matrix P  <-- P

And symbolicForm makes the basis-by-basis action of a multi-box map readable:

i23 : P = QQ[a,b,c];

i24 : symbolicForm pieri({3,1}, {1,1}, P)

      +-------+-------------+
      |+-+-+-+|  2     +-+  |
o24 = ||0|0|0||6a   *  |0|  |
      ||1| | ||        |1|  |
      |+-+-+-+|        +-+  |
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||0|0|0||6a   *  |0|  |
      ||2| | ||        |2|  |
      |+-+-+-+|        +-+  |
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||0|0|1||4a*b  *  |0| |
      ||1| | ||         |1| |
      |+-+-+-+|         +-+ |
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||0|0|1||4a*b  *  |0| |
      ||2| | ||         |2| |
      |+-+-+-+|         +-+ |
      |       |  2     +-+  |
      |       |2a   *  |1|  |
      |       |        |2|  |
      |       |        +-+  |
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||0|0|2||4a*c  *  |0| |
      ||1| | ||         |1| |
      |+-+-+-+|         +-+ |
      |       |   2     +-+ |
      |       |-2a   *  |1| |
      |       |         |2| |
      |       |         +-+ |
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||0|0|2||4a*c  *  |0| |
      ||2| | ||         |2| |
      |+-+-+-+|         +-+ |
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||0|1|1||2b   *  |0|  |
      ||1| | ||        |1|  |
      |+-+-+-+|        +-+  |
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||0|1|1||2b   *  |0|  |
      ||2| | ||        |2|  |
      |+-+-+-+|        +-+  |
      |       |         +-+ |
      |       |4a*b  *  |1| |
      |       |         |2| |
      |       |         +-+ |
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||0|1|2||2b*c  *  |0| |
      ||1| | ||         |1| |
      |+-+-+-+|         +-+ |
      |       |          +-+|
      |       |-2a*b  *  |1||
      |       |          |2||
      |       |          +-+|
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||0|1|2||2b*c  *  |0| |
      ||2| | ||         |2| |
      |+-+-+-+|         +-+ |
      |       |         +-+ |
      |       |2a*c  *  |1| |
      |       |         |2| |
      |       |         +-+ |
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||0|2|2||2c   *  |0|  |
      ||1| | ||        |1|  |
      |+-+-+-+|        +-+  |
      |       |          +-+|
      |       |-4a*c  *  |1||
      |       |          |2||
      |       |          +-+|
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||0|2|2||2c   *  |0|  |
      ||2| | ||        |2|  |
      |+-+-+-+|        +-+  |
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||1|1|1||6b   *  |1|  |
      ||2| | ||        |2|  |
      |+-+-+-+|        +-+  |
      +-------+-------------+
      |+-+-+-+|         +-+ |
      ||1|1|2||4b*c  *  |1| |
      ||2| | ||         |2| |
      |+-+-+-+|         +-+ |
      +-------+-------------+
      |+-+-+-+|  2     +-+  |
      ||1|2|2||2c   *  |1|  |
      ||2| | ||        |2|  |
      |+-+-+-+|        +-+  |
      +-------+-------------+

Ways to use pieri:

pieri(List,List,PolynomialRing)

For the programmer

The object pieri is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/PieriMaps/doc.m2:381:0.

pieri -- computes a matrix representation for a Pieri inclusion of representations of a general linear group

Description

See also

Ways to use pieri:

For the programmer