A rlosed formula for the rule of LITTLEW~OD/RICHARDSOX is given. By specialization closed formulas are gotten for the decomposition of representations of gZ( V) and pZ( V) resp. where V is a vector space (graduate vector space) over 6.
Math. Nadir. 161 (1991)
We order to this partion a YOUNG frame with p rows where the i-th row contains li boxes. The objects above mentioned are always characterized by such a partion or, equivalent, by such a YOUNG frame. For simplicity we will speak in the following only about the product or the sum of the partions and the YOUNG frames resp. and denote both partions and YOUNG frames with the same symbol (1). The product of (1) and (m) we will denote with (1) @ (m).
Shortly we recall the rule of LITTLEWOOD/RICHARDSON. We consider two YOUNQ frames (1) = (I1, . . ., I,) and (m) = (ml, . . ., mq). I n the j-th row of (m) we write in all the boxes the number j and denote these boxes by j-boxes. Now we add the boxes of (m) to (1) beginning with the 1-boxes, next the 2-boxes etc.
I n the built frame the following conditions must be satisfied : I. The frame gotten from (1) after the addition of the 1-boxes (2-boxes, . . .) must be admissible, i.e. the length of successive rows are non-increasing.
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The frame does not contain boxes with equal labels appearing in a column.
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The number of ( j + 1)-boxes is never greater than the number of j-boxes if we count up the boxes in the rows from the right to the left and from top to bottom ( j = 1, . . ., q -1).
We sum symbolically all YOUNQ frames built in this way and denote this sum as the result of the multiplication of (1) and (m).